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4d0678b4a0
- correct appearance of section environments in LyX: also non-starred sections don't have a number in tufte - use the new nohyper document class option, this way the user has full access to the PDF appearance via our PDF properties settings (thanks to the tufte developer for implementing this) This can first go to branch when the new tufte version will be released - this release will definitively happen before LyX 1.7 therefore I think it's safe to commit to trunk. git-svn-id: svn://svn.lyx.org/lyx/lyx-devel/trunk@31942 a592a061-630c-0410-9148-cb99ea01b6c8
1318 lines
23 KiB
Plaintext
1318 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
|
||
\font_osf false
|
||
\font_sf_scale 100
|
||
\font_tt_scale 100
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||
|
||
\graphics default
|
||
\paperfontsize default
|
||
\spacing single
|
||
\use_hyperref true
|
||
\pdf_bookmarks true
|
||
\pdf_bookmarksnumbered false
|
||
\pdf_bookmarksopen true
|
||
\pdf_bookmarksopenlevel 1
|
||
\pdf_breaklinks false
|
||
\pdf_pdfborder true
|
||
\pdf_colorlinks false
|
||
\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_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
|
||
\output_changes false
|
||
\author ""
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||
\author ""
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||
\end_header
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||
|
||
\begin_body
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||
|
||
\begin_layout Title
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||
Tufte Examples
|
||
\end_layout
|
||
|
||
\begin_layout Author
|
||
Fake Author
|
||
\begin_inset Note Note
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||
status open
|
||
|
||
\begin_layout Plain Layout
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||
author of this example file: Jason Waskiewicz
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||
\end_layout
|
||
|
||
\end_inset
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||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset toc
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||
LatexCommand tableofcontents
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||
|
||
\end_inset
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||
|
||
|
||
\end_layout
|
||
|
||
\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
|
||
In this document, it was endeavored to show some of the features of the
|
||
|
||
\family sans
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||
Tufte-book
|
||
\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
|
||
a calculus class is demonstrated.
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||
\end_layout
|
||
|
||
\begin_layout Section
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||
Features
|
||
\end_layout
|
||
|
||
\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
|
||
\noun default
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||
.
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||
It provides the same functionality as the
|
||
\family sans
|
||
Tufte-handout
|
||
\family default
|
||
class with book-specific additions.
|
||
Tufte's documents consist of a rather narrow column of text and a wide
|
||
column of margin notes and margin figures.
|
||
This is to improve readability.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The features provided by this format include:
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Margin figures
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Ordinary figures in text with captions in margins
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Full width figures and text when needed
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
Footnotes
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
in margins
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Limited layers of sections and subsections
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In this sample document, some of these features are demonstrated.
|
||
For a full demonstration, visit the tufte-latex website:
|
||
\begin_inset Flex URL
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
http://code.google.com/p/tufte-latex
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Page Layout
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Tufte's margins are
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
ragged right
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
rather than justified.
|
||
\begin_inset Flex Sidenote
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
To get justified text, add the option
|
||
\emph on
|
||
justified
|
||
\emph default
|
||
to the
|
||
\emph on
|
||
Custom
|
||
\emph default
|
||
field in
|
||
\family sans
|
||
Document\SpecialChar \menuseparator
|
||
Settings\SpecialChar \menuseparator
|
||
Document Class
|
||
\family default
|
||
.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
Ragged right text is used in most of his works, but the class option
|
||
\emph on
|
||
justified
|
||
\emph default
|
||
is provided to change this.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Tufte also formats his pages asymmetrically.
|
||
This means that the marginalia appear on the right side on each page, whether
|
||
odd or even.
|
||
If you prefer, you can change that and have symmetric layout, as common
|
||
in traditional book typography, by using the
|
||
\emph on
|
||
symmetric
|
||
\emph default
|
||
class option.
|
||
\begin_inset Flex Sidenote
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
To create a symmetric layout, add the option
|
||
\emph on
|
||
symmetric
|
||
\emph default
|
||
to the
|
||
\emph on
|
||
Custom
|
||
\emph default
|
||
field in
|
||
\family sans
|
||
Document\SpecialChar \menuseparator
|
||
Settings\SpecialChar \menuseparator
|
||
Document Class
|
||
\family default
|
||
.
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||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Finally, Tufte does not number his chapters or his sections.
|
||
If you like to refer to sections by number, move the
|
||
\emph on
|
||
Numbering
|
||
\emph default
|
||
slider in
|
||
\family sans
|
||
Document\SpecialChar \menuseparator
|
||
Settings\SpecialChar \menuseparator
|
||
Numbering & TOC
|
||
\family default
|
||
to the very right (as done in this document).
|
||
\begin_inset Flex Marginnote
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||
status collapsed
|
||
|
||
\begin_layout Plain Layout
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||
marginnote
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||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
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||
Figures
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Tufte uses ordinary figure floats such as the following:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
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
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||
width "100col%"
|
||
special "none"
|
||
height "1in"
|
||
height_special "totalheight"
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Imagine your favorite figure inside this box instead of this boring text.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Caption
|
||
|
||
\begin_layout Plain Layout
|
||
An ordinary figure float.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Furthermore, he uses margin figures, as shown in Fig.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
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||
reference "fig:margin-figure"
|
||
|
||
\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
|
||
width "100col%"
|
||
special "none"
|
||
height "1in"
|
||
height_special "totalheight"
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Imagine your favorite photograph of a squirrel inside this box instead of
|
||
this boring text.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Caption
|
||
|
||
\begin_layout Plain Layout
|
||
A margin figure.
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:margin-figure"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In the same vein, the
|
||
\family sans
|
||
Tufte-book
|
||
\family default
|
||
class allows the use of tables, both in the margins and in the text.
|
||
The margin tables are recommended for a small set of data to illustrate
|
||
a concept such as
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
Look, distance-time data is quadratic when the object is falling.
|
||
\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
|
||
in a full-width table instead.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Finally,
|
||
\family sans
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||
Tufte-book
|
||
\family default
|
||
provides a full-width figure.
|
||
This one takes up the entire width of the page, i.
|
||
\begin_inset space \thinspace{}
|
||
\end_inset
|
||
|
||
e., text and margin.
|
||
See Fig.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:full-width-figure"
|
||
|
||
\end_inset
|
||
|
||
for an example.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide true
|
||
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 "100line%"
|
||
special "none"
|
||
height "1in"
|
||
height_special "totalheight"
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Imagine your favorite photograph of a squirrel inside this box instead of
|
||
this boring text.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset VSpace defskip
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption
|
||
|
||
\begin_layout Plain Layout
|
||
A full-width figure.
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:full-width-figure"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
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.
|
||
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
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||
This is a sidenote that was entered using a
|
||
\family typewriter
|
||
footnote
|
||
\family default
|
||
.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
Alternatively, you can also use the
|
||
\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
|
||
\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
|
||
BibTeX
|
||
\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
|
||
BibTeX
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
Bibliography
|
||
\family default
|
||
inset inside a LyX note and use the 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 Abstract
|
||
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
|
||
width "100col%"
|
||
special "none"
|
||
height "1in"
|
||
height_special "totalheight"
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
I had a graph of the square root function here.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Caption
|
||
|
||
\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
|
||
width "100col%"
|
||
special "none"
|
||
height "1in"
|
||
height_special "totalheight"
|
||
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
|
||
|
||
\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
|
||
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
|