#LyX 2.4 created this file. For more info see https://www.lyx.org/ \lyxformat 620 \begin_document \begin_header \save_transient_properties true \origin /systemlyxdir/examples/Books/ \textclass tufte-book \use_default_options true \maintain_unincluded_children no \language english \language_package default \inputencoding utf8 \fontencoding auto \font_roman "default" "default" \font_sans "default" "default" \font_typewriter "default" "default" \font_math "auto" "auto" \font_default_family default \use_non_tex_fonts false \font_sc false \font_roman_osf false \font_sans_osf false \font_typewriter_osf false \font_sf_scale 100 100 \font_tt_scale 100 100 \use_microtype false \use_dash_ligatures false \graphics default \default_output_format pdf2 \output_sync 0 \bibtex_command bibtex \index_command default \float_placement class \float_alignment class \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 \pdf_pdfusetitle true \papersize default \use_geometry true \use_package amsmath 1 \use_package amssymb 1 \use_package cancel 1 \use_package esint 1 \use_package mathdots 1 \use_package mathtools 1 \use_package mhchem 1 \use_package stackrel 1 \use_package stmaryrd 1 \use_package undertilde 1 \cite_engine natbib \cite_engine_type authoryear \biblio_style plain \use_bibtopic false \use_indices false \paperorientation portrait \suppress_date false \justification true \use_refstyle 0 \use_minted 0 \use_lineno 0 \notefontcolor #0000ff \index Index \shortcut idx \color #008000 \end_index \secnumdepth 2 \tocdepth 2 \paragraph_separation indent \paragraph_indentation default \is_math_indent 0 \math_numbering_side default \quotes_style english \dynamic_quotes 0 \papercolumns 1 \papersides 1 \paperpagestyle default \tablestyle default \tracking_changes false \output_changes false \change_bars false \postpone_fragile_content false \html_math_output 0 \html_css_as_file 0 \html_be_strict false \docbook_table_output 0 \docbook_mathml_prefix 1 \end_header \begin_body \begin_layout Standard \begin_inset Note Note status open \begin_layout Plain Layout \series bold Note \series default : To be able to view your file as PDF you must have the \SpecialChar LaTeX -package \family sans fpl \family default installed to your \SpecialChar LaTeX system. If you are using MiK\SpecialChar TeX , you will automatically be asked to install this package when previewing your file, if you are using \SpecialChar TeX Live, use \SpecialChar TeX Live's package manager. \end_layout \begin_layout Plain Layout \series bold Note 2 \series default : If you wish to compile to PDF with plain \SpecialChar LaTeX (instead of pdf\SpecialChar TeX ), there is currently a bug in the Tufte class that causes errors. You can add the class option 'nols' (put this in the \begin_inset Quotes eld \end_inset Custom \begin_inset Quotes erd \end_inset field of Document\SpecialChar menuseparator Settings \SpecialChar menuseparator Document Class) to get rid of the errors, but this also removes letterspacing, an important part of the Tufte design. \end_layout \end_inset \end_layout \begin_layout Title Tufte Examples \end_layout \begin_layout Author Fake Author \begin_inset Note Note status open \begin_layout Plain Layout author of this example file: Jason Waskiewicz \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset CommandInset toc LatexCommand tableofcontents \end_inset \end_layout \begin_layout Chapter The Features of the Tufte-book Class \end_layout \begin_layout Standard In this document, it was endeavored to show some of the features of the \family sans Tufte-book \family default class. In the first chapter, their use is outlined. In the second chapter, their use through a handout that was created in a calculus class is demonstrated. \end_layout \begin_layout Section Features \end_layout \begin_layout Standard The \family sans Tufte-book \family default class is based on the work of \noun on Edward Tufte \noun default . 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 https://tufte-latex.github.io/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 . \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 status collapsed \begin_layout Plain Layout marginnote \end_layout \end_inset \end_layout \begin_layout Section Figures \end_layout \begin_layout Standard Tufte uses ordinary figure floats such as the following: \end_layout \begin_layout Standard \begin_inset Float figure placement document alignment document 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 Imagine your favorite figure inside this box instead of this boring text. \end_layout \end_inset \begin_inset Caption Standard \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 reference "fig:margin-figure" nolink "false" \end_inset . \end_layout \begin_layout Standard \begin_inset VSpace 5theight% \end_inset \end_layout \begin_layout Standard \begin_inset Float marginfigure placement document alignment document 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 Imagine your favorite photograph of a squirrel inside this box instead of this boring text. \end_layout \end_inset \begin_inset Caption Standard \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 \begin_inset Note Greyedout status open \begin_layout Plain Layout \series bold Note: \series default If you are using margin figures or tables you must use the default float placement option in the document settings under \family sans Float \begin_inset space ~ \end_inset Placement \family default . \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 \end_inset 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 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" nolink "false" \end_inset for an example. \end_layout \begin_layout Standard \begin_inset Float figure placement document alignment document 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 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 this boring text. \end_layout \end_inset \begin_inset VSpace defskip \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 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 status open \begin_layout Plain Layout 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 improvements 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" literal "true" \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 placement document alignment document 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 placement document alignment document 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" nolink "false" \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" nolink "false" \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 placement document alignment document 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 placement document alignment document 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