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git-svn-id: svn://svn.lyx.org/lyx/lyx-devel/trunk@31849 a592a061-630c-0410-9148-cb99ea01b6c8
1337 lines
23 KiB
Plaintext
1337 lines
23 KiB
Plaintext
#LyX 1.6.5svn created this file. For more info see http://www.lyx.org/
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\lyxformat 345
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\begin_document
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\begin_header
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\textclass tufte-book
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\use_default_options true
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\language english
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\inputencoding auto
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\font_roman default
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\font_sans default
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\font_typewriter default
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\font_default_family default
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\font_sc false
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\font_osf false
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\font_sf_scale 100
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\font_tt_scale 100
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\graphics default
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\paperfontsize default
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\spacing single
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\use_hyperref false
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\papersize default
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\use_geometry true
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\use_amsmath 1
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\use_esint 1
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\cite_engine basic
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\use_bibtopic false
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\paperorientation portrait
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\secnumdepth 2
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\tocdepth 2
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\paragraph_separation indent
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\defskip medskip
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\quotes_language english
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\papercolumns 1
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\papersides 1
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\paperpagestyle default
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\tracking_changes false
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\output_changes false
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\author ""
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\author ""
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\end_header
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\begin_body
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\begin_layout Standard
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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\series bold
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Important note:
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\series default
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The menu
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\family sans
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Insert\SpecialChar \menuseparator
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Hyperlinks
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\family default
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and the
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\family sans
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PDF properties
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\family default
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in the
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\family sans
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Document\SpecialChar \menuseparator
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Settings
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\family default
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menu cannot be used for tufte document classes!
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Title
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Tufte Examples
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\end_layout
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\begin_layout Author
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Fake Author
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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author of this example file: Jason Waskiewicz
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset CommandInset toc
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LatexCommand tableofcontents
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\end_inset
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\end_layout
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\begin_layout Chapter
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The Features of the Tufte-book Class
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\end_layout
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\begin_layout Standard
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In this document, it was endeavored to show some of the features of the
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\family sans
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Tufte-book
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\family default
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class.
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In the first chapter, their use is outlined.
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In the second chapter, their use through a handout that was created in
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a calculus class is demonstrated.
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\end_layout
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\begin_layout Section
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Features
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\end_layout
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\begin_layout Standard
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The
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\family sans
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Tufte-book
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\family default
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class is based on the work of
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\noun on
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Edward Tufte
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\noun default
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.
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It provides the same functionality as the
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\family sans
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Tufte-handout
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\family default
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class with book-specific additions.
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Tufte's documents consist of a rather narrow column of text and a wide
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column of margin notes and margin figures.
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This is to improve readability.
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\end_layout
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\begin_layout Standard
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The features provided by this format include:
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\end_layout
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\begin_layout Itemize
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Margin figures
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\end_layout
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\begin_layout Itemize
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Ordinary figures in text with captions in margins
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\end_layout
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\begin_layout Itemize
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Full width figures and text when needed
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\end_layout
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\begin_layout Itemize
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\begin_inset Quotes eld
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\end_inset
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Footnotes
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\begin_inset Quotes erd
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\end_inset
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in margins
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\end_layout
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\begin_layout Itemize
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Limited layers of sections and subsections
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\end_layout
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\begin_layout Standard
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In this sample document, some of these features are demonstrated.
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For a full demonstration, visit the tufte-latex website:
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\begin_inset Flex URL
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status collapsed
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\begin_layout Plain Layout
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http://code.google.com/p/tufte-latex
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\end_layout
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\end_inset
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.
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\end_layout
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\begin_layout Section
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Page Layout
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\end_layout
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\begin_layout Standard
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Tufte's margins are
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\begin_inset Quotes eld
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\end_inset
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ragged right
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\begin_inset Quotes erd
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\end_inset
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rather than justified.
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\begin_inset Flex Sidenote
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status open
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\begin_layout Plain Layout
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To get justified text, add the option
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\emph on
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justified
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\emph default
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to the
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\emph on
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Custom
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\emph default
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field in
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\family sans
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Document\SpecialChar \menuseparator
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Settings\SpecialChar \menuseparator
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Document Class
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\family default
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.
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\end_layout
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\end_inset
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Ragged right text is used in most of his works, but the class option
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\emph on
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justified
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\emph default
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is provided to change this.
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\end_layout
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\begin_layout Standard
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Tufte also formats his pages asymmetrically.
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This means that the marginalia appear on the right side on each page, whether
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odd or even.
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If you prefer, you can change that and have symmetric layout, as common
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in traditional book typography, by using the
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\emph on
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symmetric
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\emph default
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class option.
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\begin_inset Flex Sidenote
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status open
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\begin_layout Plain Layout
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To create a symmetric layout, add the option
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\emph on
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symmetric
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\emph default
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to the
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\emph on
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Custom
|
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\emph default
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field in
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\family sans
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Document\SpecialChar \menuseparator
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Settings\SpecialChar \menuseparator
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Document Class
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\family default
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.
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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Finally, Tufte does not number his chapters or his sections.
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If you like to refer to sections by number, move the
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\emph on
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Numbering
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\emph default
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slider in
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\family sans
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Document\SpecialChar \menuseparator
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Settings\SpecialChar \menuseparator
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Numbering & TOC
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\family default
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to the very right (as done in this document).
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\begin_inset Flex Marginnote
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status collapsed
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\begin_layout Plain Layout
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marginnote
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Section
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Figures
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\end_layout
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\begin_layout Standard
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Tufte uses ordinary figure floats such as the following:
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\end_layout
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\begin_layout Standard
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\begin_inset Float figure
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wide false
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sideways false
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status open
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\begin_layout Plain Layout
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\begin_inset Box Boxed
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position "t"
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hor_pos "c"
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has_inner_box 1
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inner_pos "t"
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use_parbox 0
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width "100col%"
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special "none"
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height "1in"
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height_special "totalheight"
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status open
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\begin_layout Plain Layout
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Imagine your favorite figure inside this box instead of this boring text.
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\end_layout
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\end_inset
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\begin_inset Caption
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\begin_layout Plain Layout
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An ordinary figure float.
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\end_layout
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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Furthermore, he uses margin figures, as shown in Fig.
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\begin_inset space ~
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\end_inset
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|
|
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:margin-figure"
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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\begin_inset Float marginfigure
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wide false
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sideways false
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status open
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\begin_layout Plain Layout
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\begin_inset Box Boxed
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position "t"
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hor_pos "c"
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has_inner_box 1
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inner_pos "t"
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use_parbox 0
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width "100col%"
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special "none"
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height "1in"
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height_special "totalheight"
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status open
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\begin_layout Plain Layout
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Imagine your favorite photograph of a squirrel inside this box instead of
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this boring text.
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\end_layout
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\end_inset
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|
\begin_inset Caption
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\begin_layout Plain Layout
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A margin figure.
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\begin_inset CommandInset label
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LatexCommand label
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name "fig:margin-figure"
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\end_inset
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\end_layout
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\end_inset
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|
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\end_layout
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\end_inset
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|
|
|
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\end_layout
|
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|
|
\begin_layout Standard
|
|
In the same vein, the
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\family sans
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Tufte-book
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\family default
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class allows the use of tables, both in the margins and in the text.
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The margin tables are recommended for a small set of data to illustrate
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a concept such as
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\begin_inset Quotes eld
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\end_inset
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Look, distance-time data is quadratic when the object is falling.
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\begin_inset Quotes erd
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\end_inset
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More detailed data such as flame test results and comments should be put
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in a full-width table instead.
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\end_layout
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\begin_layout Standard
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Finally,
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\family sans
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Tufte-book
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\family default
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provides a full-width figure.
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This one takes up the entire width of the page, i.
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\begin_inset space \thinspace{}
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\end_inset
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e., text and margin.
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See Fig.
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\begin_inset space ~
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\end_inset
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|
|
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:full-width-figure"
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\end_inset
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for an example.
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\end_layout
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\begin_layout Standard
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\begin_inset Float figure
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wide true
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sideways false
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status open
|
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|
\begin_layout Plain Layout
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\begin_inset Box Boxed
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position "t"
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hor_pos "c"
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has_inner_box 1
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inner_pos "t"
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use_parbox 0
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width "100col%"
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special "none"
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height "1in"
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height_special "totalheight"
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status open
|
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|
|
\begin_layout Plain Layout
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Imagine a photograph of a squirrel stretched out on its side in this box.
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One of the bugs in my layout is that this only works with pictures, not
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with frames around minipages, the way it's set up here.
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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\begin_inset Caption
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\begin_layout Plain Layout
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A full-width figure.
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\begin_inset CommandInset label
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LatexCommand label
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name "fig:full-width-figure"
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\end_inset
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|
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\end_layout
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\begin_layout Section
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Working with Text
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\end_layout
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\begin_layout Standard
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One of the most prominent and distinctive features of this style is the
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extensive use of sidenotes.
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There is a wide margin to provide ample room for sidenotes and small figures.
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Any footnotes will automatically be converted to sidenotes.
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\begin_inset Foot
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status open
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\begin_layout Plain Layout
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This is a sidenote that was entered using a
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\family typewriter
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footnote
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\family default
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.
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\end_layout
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|
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\end_inset
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Alternatively, you can also use the
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\family typewriter
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Sidenote
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\family default
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inset directly; you'll find it in the
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\family sans
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Insert\SpecialChar \menuseparator
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Custom Insets
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\family default
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menu.
|
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\begin_inset Flex Sidenote
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status open
|
|
|
|
\begin_layout Plain Layout
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|
This is a sidenote that was entered using a
|
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\family typewriter
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sidenote
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\family default
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.
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\end_layout
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\end_inset
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|
|
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|
\end_layout
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|
|
\begin_layout Standard
|
|
If you like to place ancillary information in the margin without the sidenote
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mark (the superscript number), you can use the
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\family typewriter
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Marginnote
|
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\family default
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inset.
|
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|
|
\begin_inset Flex Marginnote
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status open
|
|
|
|
\begin_layout Plain Layout
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This is Tufte's margin note.
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Notice that there isn't a number preceding the note, and there is no number
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in the main text where this note was written.
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\end_layout
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|
\end_inset
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|
The normal margin note will work as well, but it will look rather odd.
|
|
\begin_inset Marginal
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status open
|
|
|
|
\begin_layout Plain Layout
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This is a normal margin note.
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Don't use it.
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\end_layout
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|
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|
\end_inset
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|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
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|
\begin_inset Flex NewThought
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status open
|
|
|
|
\begin_layout Plain Layout
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Another useful
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\end_layout
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|
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|
\end_inset
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|
|
|
innovation is Tufte's
|
|
\family typewriter
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|
NewThought
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\family default
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|
character style (
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\family sans
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|
Edit\SpecialChar \menuseparator
|
|
TextStyle\SpecialChar \menuseparator
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|
NewThought
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\family default
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|
).
|
|
It introduces new thoughts by means of small caps, as demonstrated in this
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paragraph.
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|
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|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The Tufte document classes include two new character styles and some improvement
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s on existing commands for letterspacing.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
When setting strings of
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|
\begin_inset Flex AllCaps
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|
status open
|
|
|
|
\begin_layout Plain Layout
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|
ALL CAPS
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\end_layout
|
|
|
|
\end_inset
|
|
|
|
or
|
|
\begin_inset Flex SmallCaps
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|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Small Caps
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
, the letterspacing---that is, the spacing between the letters---should
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|
be increased slightly.
|
|
\begin_inset CommandInset citation
|
|
LatexCommand cite
|
|
key "Bringhurst2005"
|
|
|
|
\end_inset
|
|
|
|
The
|
|
\emph on
|
|
AllCaps
|
|
\emph default
|
|
character style (
|
|
\family sans
|
|
Edit\SpecialChar \menuseparator
|
|
TextStyle\SpecialChar \menuseparator
|
|
AllCaps
|
|
\family default
|
|
) has proper letterspacing for strings of
|
|
\begin_inset Flex AllCaps
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
FULL CAPITAL LETTERS
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
, and the
|
|
\emph on
|
|
SmallCaps
|
|
\emph default
|
|
character style (
|
|
\family sans
|
|
Edit\SpecialChar \menuseparator
|
|
TextStyle\SpecialChar \menuseparator
|
|
SmallCaps
|
|
\family default
|
|
) has letterspacing for
|
|
\begin_inset Flex SmallCaps
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
small capital letters
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
.
|
|
These commands will also automatically convert the case of the text to
|
|
upper- or lowercase, respectively.
|
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\end_layout
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\begin_layout Standard
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The normal
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\emph on
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Small Caps
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\emph default
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shape has also been redefined to include letterspacing.
|
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Its case is left as is, however.
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This allows one to use both uppercase and lowercase letters:
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\shape smallcaps
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The Initial Letters Of The Words In This Sentence Are Capitalized.
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\end_layout
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\begin_layout Standard
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Finally, the
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\emph on
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Full Width
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\emph default
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paragraph environment provides a paragraph layout that stretches across
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the main text block and the sidenotes area:
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\end_layout
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\begin_layout Full Width
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Lorem ipsum dolor sit amet, consectetuer adipiscing elit.
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Ut purus elit, vestibulum ut, placerat ac, adipiscing vitae, felis.
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Curabitur dictum gravida mauris.
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Nam arcu libero, nonummy eget, consectetuer id, vulputate a, magna.
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Donec vehicula augue eu neque.
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Pellentesque habitant morbi tristique senectus et netus et malesuada fames
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ac turpis egestas.
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Mauris ut leo.
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Cras viverra metus rhoncus sem.
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Nulla et lectus vestibulum urna fringilla ultrices.
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Phasellus eu tellus sit amet tortor gravida placerat.
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Integer sapien est, iaculis in, pretium quis, viverra ac, nunc.
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Praesent eget sem vel leo ultrices bibendum.
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Aenean faucibus.
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Morbi dolor nulla, malesuada eu, pulvinar at, mollis ac, nulla.
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Donec varius orci eget risus.
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Duis nibh mi, congue eu, accumsan eleifend, sagittis quis, diam.
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Duis eget orci sit amet orci dignissim rutrum.
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\end_layout
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\begin_layout Section
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References
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\end_layout
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\begin_layout Standard
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References are placed alongside their citations as sidenotes, as well.
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This can be accomplished using the normal citation command.
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\begin_inset Foot
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status collapsed
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\begin_layout Plain Layout
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The previous section includes a citation.
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\end_inset
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\end_layout
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The complete list of references may also be printed automatically by using
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\family sans
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Insert\SpecialChar \menuseparator
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Lists/TOC\SpecialChar \menuseparator
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BibTeX
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\begin_inset space ~
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\end_inset
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Bibliography
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\family default
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(see the end of this document for an example.) If you do not want to print
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a bibliography at the end of your document, place the
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\family sans
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BibTeX
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\begin_inset space ~
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\end_inset
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Bibliography
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inset inside a LyX note and use the LaTeX command
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\family typewriter
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\backslash
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nobibliography
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the text.
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\end_layout
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\begin_layout Chapter
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Calculation of Volume: Sections 2.12--2.13
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\end_layout
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\begin_layout Abstract
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Imagine taking a function like
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\begin_inset Formula $y=\sqrt{x}$
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\end_inset
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|
|
|
and rotating it in 3 dimensions around the x-axis.
|
|
The resulting shape would look somewhat like a cup (on its side).
|
|
Interestingly, integration empowers us to do exactly this and to find out
|
|
how much water that cup could hold.
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\end_layout
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\begin_layout Section
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Visualizing Rotation
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\end_layout
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\begin_layout Standard
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\begin_inset Float marginfigure
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wide false
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sideways false
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status open
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\begin_layout Plain Layout
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\begin_inset Box Boxed
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position "t"
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width "100col%"
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height "1in"
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height_special "totalheight"
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status open
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\begin_layout Plain Layout
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I had a graph of the square root function here.
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\end_inset
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\begin_inset Caption
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\begin_layout Plain Layout
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LatexCommand label
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name "mar:A-graph-of"
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\end_inset
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A graph of
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\begin_inset Formula $f(x)=\sqrt{x}$
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\end_inset
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\end_layout
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\end_inset
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\begin_inset Float marginfigure
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sideways false
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position "t"
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hor_pos "c"
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has_inner_box 1
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inner_pos "t"
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use_parbox 0
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width "100col%"
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special "none"
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height "1in"
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height_special "totalheight"
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status open
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\begin_layout Plain Layout
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Here I rotated the square root function and then drew a disk on the figure
|
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to illustrate how I would calculate the volume of the figure.
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\end_layout
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\end_inset
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\begin_inset Caption
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\begin_layout Plain Layout
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\begin_inset CommandInset label
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LatexCommand label
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name "mar:rotated"
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\end_inset
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A graph of
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\begin_inset Formula $f(x)=\sqrt{x}$
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\end_inset
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rotated about the x-axis and with additional remarks for integration.
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\end_layout
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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Figure
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "mar:A-graph-of"
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|
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\end_inset
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|
|
shows the plot of the function
|
|
\begin_inset Formula $f(x)=\sqrt{x}.$
|
|
\end_inset
|
|
|
|
Now, imagine that we rotate that function about the x-axis.
|
|
The resulting figure would be somewhat like figure
|
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\begin_inset CommandInset ref
|
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LatexCommand ref
|
|
reference "mar:rotated"
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|
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\end_inset
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|
|
.
|
|
This is akin to a cup lying on its side.
|
|
For the sake of clarity, the artist (me) drew a circle on the end of the
|
|
figure to show that it is indeed rotated.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Now, suppose we wished to find the volume of the figure.
|
|
When we integrated the original square root function to find its area,
|
|
we imagined a series of rectangles inside the figure.
|
|
Their height was
|
|
\begin_inset Formula $h=f(x)$
|
|
\end_inset
|
|
|
|
and their width was
|
|
\begin_inset Formula $dx$
|
|
\end_inset
|
|
|
|
.
|
|
Since height multiplied by width was the area of each rectangle, we summed
|
|
these areas and rewrote this as
|
|
\begin_inset Formula $\int\, f(x)\, dx$
|
|
\end_inset
|
|
|
|
, or, in this specific case,
|
|
\begin_inset Formula $\int\,\sqrt{x}\, dx$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
To find the volume of our rotated figure the prodecure is quite similar.
|
|
Begin by rotating each rectangle about the x-axis.
|
|
This creates a series of cylinders.
|
|
\begin_inset Flex Sidenote
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
The text refers to these cylinders as "disks".
|
|
This is standard practice in all the Calculus books I checked.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
Then, we can find the volume of each cylinder/disk.
|
|
The basic formula is:
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \[
|
|
V=hA\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\noindent
|
|
where
|
|
\begin_inset Formula $h$
|
|
\end_inset
|
|
|
|
is the height of the cylinder (width of the rectangle)
|
|
\begin_inset Formula $dx$
|
|
\end_inset
|
|
|
|
.
|
|
The area of each figure is a circle where
|
|
\begin_inset Formula $A=\pi r^{2}$
|
|
\end_inset
|
|
|
|
.
|
|
The radius in this case is the function
|
|
\begin_inset Formula $f(x)$
|
|
\end_inset
|
|
|
|
.
|
|
By substitution
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \[
|
|
A=\pi f^{2}(x)\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\noindent
|
|
In our specific case,
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \begin{eqnarray*}
|
|
A & = & \pi\left(\sqrt{x}\right)^{2}\\
|
|
& = & \pi x\end{eqnarray*}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
To calculate the volume of one disk, we have
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \[
|
|
V=\pi x\, dx\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\noindent
|
|
or, in the general case
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \[
|
|
V=\pi f^{2}(x)\, dx\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
To find the volume of the figure between points
|
|
\begin_inset Formula $a$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $b$
|
|
\end_inset
|
|
|
|
we sum the volumes by means of integration:
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \begin{equation}
|
|
\int_{a}^{b}\,\pi f^{2}(x)\, dx\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\noindent
|
|
In the specific example, over the interval
|
|
\begin_inset Formula $[0,4]$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \begin{eqnarray*}
|
|
\int_{0}^{4}\,\pi x\, dx & = & \pi\int_{0}^{4}\, x\, dx\\
|
|
& = & \pi\left.\left(\frac{x^{2}}{2}\right)\right|_{0}^{4}\\
|
|
& = & \pi\left(\frac{4^{2}}{2}-0\right)\\
|
|
& = & 8\pi\end{eqnarray*}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Flex NewThought
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Suppose I drill
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
a special hole down the length of the cup we just worked with.
|
|
It is made with a quadratic shaped bit.
|
|
\begin_inset Flex Sidenote
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
I have no idea how I'd do this in real life, but I'm making a point.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
I find that the hole the bit makes can be modeled with the function
|
|
\begin_inset Formula $g(x)=\frac{x^{2}}{16}$
|
|
\end_inset
|
|
|
|
.
|
|
I would need to subtract the volume of the material removed from the volume
|
|
of the entire
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
cup
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
.
|
|
Each individual cylinder would become like a
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
washer
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
.
|
|
To get the area of one washer, I would use the formula
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float marginfigure
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Box Boxed
|
|
position "t"
|
|
hor_pos "c"
|
|
has_inner_box 1
|
|
inner_pos "t"
|
|
use_parbox 0
|
|
width "100col%"
|
|
special "none"
|
|
height "1in"
|
|
height_special "totalheight"
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Here I used RLPlot to draw the square root function and the quadratic function.
|
|
Then I used Inkscape to shade the area between them.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Caption
|
|
|
|
\begin_layout Plain Layout
|
|
The functions
|
|
\begin_inset Formula $f(x)$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $g(x)$
|
|
\end_inset
|
|
|
|
and the area left by
|
|
\begin_inset Formula $f(x)-g(x)$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \begin{eqnarray*}
|
|
A_{washer} & = & A_{cup}-A_{drill}\\
|
|
& = & \pi f^{2}(x)-\pi g^{2}(x)\\
|
|
& = & \pi\left(f^{2}(x)-g^{2}(x)\right)\end{eqnarray*}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The volume of each washer would be
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \begin{eqnarray*}
|
|
V_{washer} & = & A_{washer}\, dx\\
|
|
& = & \pi\left(f^{2}(x)-g^{2}(x)\right)\, dx\end{eqnarray*}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\noindent
|
|
Then, by summing the volumes of all the washers between points
|
|
\begin_inset Formula $a$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $b$
|
|
\end_inset
|
|
|
|
, the integral is derived:
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \begin{equation}
|
|
\int_{a}^{b}\,\pi\left(f^{2}(x)-g^{2}(x)\right)\, dx\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
In the case of our quadratic drill bit::
|
|
\begin_inset Float marginfigure
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Box Boxed
|
|
position "t"
|
|
hor_pos "c"
|
|
has_inner_box 1
|
|
inner_pos "t"
|
|
use_parbox 0
|
|
width "100col%"
|
|
special "none"
|
|
height "1in"
|
|
height_special "totalheight"
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
This was the rotated set of 2 functions.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Caption
|
|
|
|
\begin_layout Plain Layout
|
|
The cup with a quadratic hole drilled down its length
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula \begin{eqnarray*}
|
|
\int_{0}^{4}\,\pi\left(\left(\sqrt{x}\right)^{2}-\left(\frac{x^{2}}{16}\right)\right)\, dx & = & \pi\int_{0}^{4}\,\left(x-\frac{x^{4}}{256}\right)\, dx\\
|
|
& = & \pi\left(\int_{0}^{4}\, x\, dx-\int_{0}^{4}\,\frac{x^{4}}{256}\, dx\right)\\
|
|
& = & \pi\left(\left.\left(\frac{x^{2}}{2}\right)\right|_{0}^{4}-\left.\left(\frac{x^{5}}{1280}\right)\right|_{0}^{4}\right)\\
|
|
& = & \pi\left(\left(\frac{4^{2}}{2}-0\right)-\left(\frac{4^{5}}{1280}-0\right)\right)\\
|
|
& = & \pi\left(8-0.8\right)\\
|
|
& = & 7.2\pi\end{eqnarray*}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
Homework
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
p114: 1, 4, 5, 6, 8, 10, 12, 15
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset CommandInset bibtex
|
|
LatexCommand bibtex
|
|
bibfiles "biblioExample"
|
|
options "plainnat"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_body
|
|
\end_document
|