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1500 lines
26 KiB
Plaintext
1500 lines
26 KiB
Plaintext
#LyX 2.2 created this file. For more info see http://www.lyx.org/
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\lyxformat 508
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\begin_document
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\begin_header
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\save_transient_properties true
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\origin /systemlyxdir/examples/
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\textclass tufte-book
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\use_default_options true
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\maintain_unincluded_children false
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\language english
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\language_package default
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\inputencoding auto
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\fontencoding global
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||
\font_roman "default" "default"
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||
\font_sans "default" "default"
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||
\font_typewriter "default" "default"
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||
\font_math "auto" "auto"
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||
\font_default_family default
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||
\use_non_tex_fonts false
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||
\font_sc false
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||
\font_osf false
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||
\font_sf_scale 100 100
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||
\font_tt_scale 100 100
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||
\graphics default
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||
\default_output_format pdf2
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||
\output_sync 0
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||
\bibtex_command bibtex
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\index_command default
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||
\paperfontsize default
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\spacing single
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\use_hyperref true
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||
\pdf_bookmarks true
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||
\pdf_bookmarksnumbered false
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\pdf_bookmarksopen true
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\pdf_bookmarksopenlevel 1
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\pdf_breaklinks false
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\pdf_pdfborder true
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||
\pdf_colorlinks false
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||
\pdf_backref false
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\pdf_pdfusetitle true
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\papersize default
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||
\use_geometry true
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\use_package amsmath 1
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\use_package amssymb 1
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||
\use_package cancel 1
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||
\use_package esint 1
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||
\use_package mathdots 1
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\use_package mathtools 1
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\use_package mhchem 1
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\use_package stackrel 1
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\use_package stmaryrd 1
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||
\use_package undertilde 1
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\cite_engine basic
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||
\cite_engine_type default
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||
\biblio_style plain
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||
\use_bibtopic false
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||
\use_indices false
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||
\paperorientation portrait
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||
\suppress_date false
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\justification true
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\use_refstyle 0
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\notefontcolor #0000ff
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\index Index
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\shortcut idx
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\color #008000
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\end_index
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\secnumdepth 2
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\tocdepth 2
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\paragraph_separation indent
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\paragraph_indentation default
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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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\html_math_output 0
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\html_css_as_file 0
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\html_be_strict false
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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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Note
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\series default
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: To be able to view your file as PDF you must have the \SpecialChar LaTeX
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-package
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\family sans
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fpl
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\family default
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installed to your \SpecialChar LaTeX
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system.
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If you are using MiK\SpecialChar TeX
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, you will automatically be asked to install this package
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when previewing your file, if you are using \SpecialChar TeX
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Live, use \SpecialChar TeX
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Live's package manager.
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\end_layout
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\begin_layout Plain Layout
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\series bold
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Note 2
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\series default
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: If you wish to compile to PDF with plain \SpecialChar LaTeX
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(instead of pdf\SpecialChar TeX
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), there is currently
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a bug in the Tufte class that causes errors.
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You can add the class option 'nols' (put this in the
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\begin_inset Quotes eld
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\end_inset
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Custom
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\begin_inset Quotes erd
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\end_inset
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field of Document\SpecialChar menuseparator
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Settings \SpecialChar menuseparator
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Document Class) to get rid of the errors, but
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this also removes letterspacing, an important part of the Tufte design.
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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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https://tufte-latex.github.io/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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use_makebox 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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thickness "0.4pt"
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separation "3pt"
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shadowsize "4pt"
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framecolor "black"
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backgroundcolor "none"
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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 Standard
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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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\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 VSpace 5theight%
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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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||
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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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use_makebox 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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thickness "0.4pt"
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separation "3pt"
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shadowsize "4pt"
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framecolor "black"
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backgroundcolor "none"
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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 Standard
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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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\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
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\begin_inset Note Greyedout
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status open
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\begin_layout Plain Layout
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||
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\series bold
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||
Note:
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\series default
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If you are using margin figures or tables you must use the default float
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placement option in the document settings under
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\family sans
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Float
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\begin_inset space ~
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\end_inset
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Placement
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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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\end_layout
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\begin_layout Standard
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||
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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||
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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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||
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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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||
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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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||
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||
\end_inset
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||
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||
for an example.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
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||
wide true
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||
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
|
||
use_makebox 0
|
||
width "100line%"
|
||
special "none"
|
||
height "1in"
|
||
height_special "totalheight"
|
||
thickness "0.4pt"
|
||
separation "3pt"
|
||
shadowsize "4pt"
|
||
framecolor "black"
|
||
backgroundcolor "none"
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Imagine your favorite photograph of a squirrel inside this box instead of
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||
this boring text.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset VSpace defskip
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||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
A full-width figure.
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
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||
name "fig:full-width-figure"
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||
|
||
\end_inset
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||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
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||
Working with Text
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
One of the most prominent and distinctive features of this style is the
|
||
extensive use of sidenotes.
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||
There is a wide margin to provide ample room for sidenotes and small figures.
|
||
Any footnotes will automatically be converted to sidenotes.
|
||
\begin_inset Foot
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||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
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
|
||
|
||
\end_inset
|
||
|
||
Alternatively, you can also use the
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||
\family typewriter
|
||
Sidenote
|
||
\family default
|
||
inset directly; you'll find it in the
|
||
\family sans
|
||
Insert\SpecialChar menuseparator
|
||
Custom Insets
|
||
\family default
|
||
menu.
|
||
\begin_inset Flex Sidenote
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
This is a sidenote that was entered using a
|
||
\family typewriter
|
||
sidenote
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||
\family default
|
||
.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
If you like to place ancillary information in the margin without the sidenote
|
||
mark (the superscript number), you can use the
|
||
\family typewriter
|
||
Marginnote
|
||
\family default
|
||
inset.
|
||
|
||
\begin_inset Flex Marginnote
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
This is Tufte's margin note.
|
||
Notice that there isn't a number preceding the note, and there is no number
|
||
in the main text where this note was written.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
The normal margin note will work as well, but it will look rather odd.
|
||
\begin_inset Marginal
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
This is a normal margin note.
|
||
Don't use it.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Flex NewThought
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Another useful
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
innovation is Tufte's
|
||
\family typewriter
|
||
NewThought
|
||
\family default
|
||
character style (
|
||
\family sans
|
||
Edit\SpecialChar menuseparator
|
||
TextStyle\SpecialChar menuseparator
|
||
NewThought
|
||
\family default
|
||
).
|
||
It introduces new thoughts by means of small caps, as demonstrated in this
|
||
paragraph.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The Tufte document classes include two new character styles and some improvement
|
||
s on existing commands for letterspacing.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
When setting strings of
|
||
\begin_inset Flex AllCaps
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
ALL CAPS
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
or
|
||
\begin_inset Flex SmallCaps
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
small caps
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
, the letterspacing—that is, the spacing between the letters—should 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.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The normal
|
||
\emph on
|
||
Small Caps
|
||
\emph default
|
||
shape has also been redefined to include letterspacing.
|
||
Its case is left as is, however.
|
||
This allows one to use both uppercase and lowercase letters:
|
||
\shape smallcaps
|
||
The Initial Letters Of The Words In This Sentence Are Capitalized.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Finally, the
|
||
\emph on
|
||
Full Width
|
||
\emph default
|
||
paragraph environment provides a paragraph layout that stretches across
|
||
the main text block and the sidenotes area:
|
||
\end_layout
|
||
|
||
\begin_layout Full Width
|
||
Lorem ipsum dolor sit amet, consectetuer adipiscing elit.
|
||
Ut purus elit, vestibulum ut, placerat ac, adipiscing vitae, felis.
|
||
Curabitur dictum gravida mauris.
|
||
Nam arcu libero, nonummy eget, consectetuer id, vulputate a, magna.
|
||
Donec vehicula augue eu neque.
|
||
Pellentesque habitant morbi tristique senectus et netus et malesuada fames
|
||
ac turpis egestas.
|
||
Mauris ut leo.
|
||
Cras viverra metus rhoncus sem.
|
||
Nulla et lectus vestibulum urna fringilla ultrices.
|
||
Phasellus eu tellus sit amet tortor gravida placerat.
|
||
Integer sapien est, iaculis in, pretium quis, viverra ac, nunc.
|
||
Praesent eget sem vel leo ultrices bibendum.
|
||
Aenean faucibus.
|
||
Morbi dolor nulla, malesuada eu, pulvinar at, mollis ac, nulla.
|
||
Curabitur auctor semper nulla.
|
||
Donec varius orci eget risus.
|
||
Duis nibh mi, congue eu, accumsan eleifend, sagittis quis, diam.
|
||
Duis eget orci sit amet orci dignissim rutrum.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
References
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
References are placed alongside their citations as sidenotes, as well.
|
||
This can be accomplished using the normal citation command.
|
||
\begin_inset Foot
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
The previous section includes a citation.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The complete list of references may also be printed automatically by using
|
||
|
||
\family sans
|
||
Insert\SpecialChar menuseparator
|
||
Lists/TOC\SpecialChar menuseparator
|
||
Bib\SpecialChar TeX
|
||
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
Bibliography
|
||
\family default
|
||
(see the end of this document for an example.) If you do not want to print
|
||
a bibliography at the end of your document, place the
|
||
\family sans
|
||
Bib\SpecialChar TeX
|
||
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
Bibliography
|
||
\family default
|
||
inset inside a \SpecialChar LyX
|
||
note and use the \SpecialChar LaTeX
|
||
command
|
||
\family typewriter
|
||
|
||
\backslash
|
||
nobibliography
|
||
\family default
|
||
the text.
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Calculation of Volume: Sections 2.12–2.13
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Imagine taking a function like
|
||
\begin_inset Formula $y=\sqrt{x}$
|
||
\end_inset
|
||
|
||
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.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Visualizing Rotation
|
||
\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
|
||
use_makebox 0
|
||
width "100col%"
|
||
special "none"
|
||
height "1in"
|
||
height_special "totalheight"
|
||
thickness "0.4pt"
|
||
separation "3pt"
|
||
shadowsize "4pt"
|
||
framecolor "black"
|
||
backgroundcolor "none"
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
I had a graph of the square root function here.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "mar:A-graph-of"
|
||
|
||
\end_inset
|
||
|
||
A graph of
|
||
\begin_inset Formula $f(x)=\sqrt{x}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\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
|
||
use_makebox 0
|
||
width "100col%"
|
||
special "none"
|
||
height "1in"
|
||
height_special "totalheight"
|
||
thickness "0.4pt"
|
||
separation "3pt"
|
||
shadowsize "4pt"
|
||
framecolor "black"
|
||
backgroundcolor "none"
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Here I rotated the square root function and then drew a disk on the figure
|
||
to illustrate how I would calculate the volume of the figure.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "mar:rotated"
|
||
|
||
\end_inset
|
||
|
||
A graph of
|
||
\begin_inset Formula $f(x)=\sqrt{x}$
|
||
\end_inset
|
||
|
||
rotated about the x-axis and with additional remarks for integration.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "mar:A-graph-of"
|
||
|
||
\end_inset
|
||
|
||
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
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "mar:rotated"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
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
|
||
use_makebox 0
|
||
width "100col%"
|
||
special "none"
|
||
height "1in"
|
||
height_special "totalheight"
|
||
thickness "0.4pt"
|
||
separation "3pt"
|
||
shadowsize "4pt"
|
||
framecolor "black"
|
||
backgroundcolor "none"
|
||
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 Standard
|
||
|
||
\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
|
||
use_makebox 0
|
||
width "100col%"
|
||
special "none"
|
||
height "1in"
|
||
height_special "totalheight"
|
||
thickness "0.4pt"
|
||
separation "3pt"
|
||
shadowsize "4pt"
|
||
framecolor "black"
|
||
backgroundcolor "none"
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
This was the rotated set of 2 functions.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Caption Standard
|
||
|
||
\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
|