\thepage } %} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \subsection*{Ma 227 \hspace{2.25in} Exam IIA \ \ Solutions\hfill 4/5/04} \subsection*{{\protect\normalsize Name: \protect\underline{\hspace{2.5in}} \hfill ID: \protect\underline{\hspace{2.0in}}}} \subsection*{{\protect\normalsize Lecture Section: \_\_\_\_\_\_\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad\ \ \ \ \ \ \ \ \ \ \ Recitation Section: \_\_\_\_\hspace{2.5in}\hfill }} \hfill {\small \textit{I pledge my honor that I have abided by the Stevens Honor System.}\hspace{2pt} \ \ \ \ \ \ \ \ \ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_% \_\_\_\_} \hfill \paragraph{You may not use a calculator, cell phone, or computer while taking this exam. All work must be shown to obtain full credit. Credit will not be given for work not reasonably supported. When you finish, be sure to sign the pledge.} \vspace{1pt} Score on Problem \ \#1 \_\_\_\_\_\_\_\_\vspace{0.2cm} \qquad \qquad \qquad \qquad\ \ \ \#2 \_\_\_\_\_\_\_\_\vspace{0.2cm} \qquad \qquad \qquad \qquad\ \ \ \#3 \_\_\_\_\_\_\_\_\vspace{0.2cm} \qquad \qquad \qquad \qquad\ \ \ \#4 \_\_\_\_\_\_\_\_ \qquad \qquad \qquad \qquad\ \ \qquad \qquad \qquad \qquad\ \ \qquad \qquad \qquad \qquad\ \ \qquad \qquad \qquad \qquad\ \ \qquad \qquad \qquad \qquad\ \ Total Score \ \ \ \ \ \ \ \ \ \ \ \ \ \ \_\_\_\_\_\_\_\_\bigskip \pagebreak \begin{description} \item[{1 [$25$ pts.]}] Give two expressions with two different orders of integration for \begin{equation*} \diint\limits_{R}xydA \end{equation*} \end{description} where $R$ is the region bounded by the lines $y=0,y=2x,$ and $x=2.$ Be sure to sketch $R.$ Evaluate \textit{one} of these expressions. Solution: $R$ is the triangular region shown below. $x$\FRAME{dtbpFX}{4.4996in}{3in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 4.4996in;height 3in;depth 0pt;display "USEDEF";plot_snapshots FALSE;mustRecompute FALSE;lastEngine "Maple";xmin "0";xmax "2";xviewmin "-0.04";xviewmax "2.0408";yviewmin "-0.08";yviewmax "4.0816";plottype 4;constrained TRUE;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$2x$};linecolor "black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "0,2";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function \TEXUX{$\left( 2,0,2,4\right) $};linecolor "black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";curveColor "[flat::RGB:0000000000]";curveStyle "Line";function \TEXUX{$\left( 0,0,2,0\right) $};linecolor "black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";curveColor "[flat::RGB:0000000000]";curveStyle "Line";}} When $x=2,$ then $y=4.$ Therefore% \begin{equation*} \int_{0}^{4}\int_{\frac{y}{2}}^{2}xydxdy=\int_{0}^{4}\left. \frac{x^{2}}{2}% \right] _{\frac{y}{2}}^{2}ydy=\int_{0}^{4}\left( 2y-\frac{y^{3}}{8}\right) dy=\left. y^{2}-\frac{y^{4}}{32}\right] _{0}^{4}=16-8=8 \end{equation*} or% \begin{equation*} \int_{0}^{2}\int_{0}^{2x}xydydx=\int_{0}^{2}\left. \frac{y^{2}}{2}\right] _{0}^{2x}xdx=\int_{0}^{2}2x^{3}dx=\left. \frac{x^{4}}{2}\right] _{0}^{2}=8 \end{equation*} \pagebreak \begin{description} \item[{2 a $\left[ 20\text{ pts.}\right] $}] Give two different triple integral expressions for the volume of the region beneath the graph with equation \end{description} $z=\frac{1}{2}x^{2}+\frac{1}{3}y^{2}$ and above the triangular region $R$ in the $x,y-$ plane with vertices $\left( 0,0,0\right) ,\left( 1,1,0\right) ,\left( 0,2,0\right) .$ Be sure to sketch $R.$ \textit{Do not evaluate these expressions.} Solution: The region $R$ is shown below. $\left( 0,0,1,1,1,1,0,2\right) $\FRAME{dtbpFX}{4.4996in}{3in}{0pt}{}{}{Plot}{% \special{language "Scientific Word";type "MAPLEPLOT";width 4.4996in;height 3in;depth 0pt;display "USEDEF";plot_snapshots FALSE;mustRecompute FALSE;lastEngine "Maple";xmin "-5";xmax "5";xviewmin "-0.02";xviewmax "1.0204";yviewmin "-0.04";yviewmax "2.0408";plottype 4;constrained TRUE;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\left( 0,0,1,1,1,1,0,2,0,0\right) $};linecolor "black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";curveColor "[flat::RGB:0000000000]";curveStyle "Line";}} We need the equations of the lines that bound $R.$ The line through $\left( 0,0\right) $ and $\left( 1,1\right) $ is $y=x.$ The line through $\left( 0,2\right) $ and $\left( 1,1\right) $ is $y=-x+2.$ Therefore% \begin{eqnarray*} \text{volume} &=&\int_{0}^{1}\int_{x}^{-x+2}\int_{0}^{\frac{1}{2}x^{2}+\frac{% 1}{3}y^{2}}dzdydx \\ &=&\int_{0}^{1}\int_{0}^{y}\int_{0}^{\frac{1}{2}x^{2}+\frac{1}{3}% y^{2}}dzdxdy+\int_{1}^{2}\int_{0}^{-y+2}\int_{0}^{\frac{1}{2}x^{2}+\frac{1}{3% }y^{2}}dzdxdy \end{eqnarray*} \begin{description} \item[{2 b $\left[ 20\text{ pts.}\right] $}] Consider the integral% \begin{equation*} \int_{0}^{2}\int_{0}^{\sqrt{2x-x^{2}}}y\left( x^{2}+y^{2}\right) ^{\frac{1}{2% }}dydx \end{equation*} \end{description} Sketch the region of integration and then convert the integral to polar coordinates. \textit{Do} \textit{not evaluate.} Solution: Note that $y$ goes from $0$ to $\sqrt{2x-x^{2}}.$ Thus $y=\sqrt{% 2x-x^{2}}$ or $y^{2}=2x-x^{2}$ so we have the upper half of the circle% \begin{equation*} x^{2}+y^{2}-2x=0 \end{equation*} or% \begin{equation*} \left( x-1\right) ^{2}+y^{2}=1 \end{equation*} which is the upper half of the circle of radius $1$ with center at $\left( 1,0\right) $. This region is shown below. $\sqrt{2x-x^{2}}$\FRAME{dtbpFX}{4.4998in}{2.9999in}{0pt}{}{}{Plot}{\special% {language "Scientific Word";type "MAPLEPLOT";width 4.4998in;height 2.9999in;depth 0pt;display "USEDEF";plot_snapshots FALSE;mustRecompute FALSE;lastEngine "Maple";xmin "0";xmax "2";xviewmin "-0.04";xviewmax "2.0408";yviewmin "-0.019998866742912E0";yviewmax "1.02034218122338";plottype 4;constrained TRUE;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};var1name \TEXUX{$x$};function \TEXUX{$\sqrt{2x-x^{2}}$};linecolor "black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "0,2";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function \TEXUX{$\left( 0,0,2,0\right) $};linecolor "black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";curveColor "[flat::RGB:0000000000]";curveStyle "Line";}} In polar coordinates the equation of the circle is% \begin{equation*} x^{2}+y^{2}=r^{2}=2x=2r\cos \theta \end{equation*} or% \begin{equation*} r=2\cos \theta \end{equation*} Thus% \begin{equation*} \int_{0}^{2}\int_{0}^{\sqrt{2x-x^{2}}}y\left( x^{2}+y^{2}\right) ^{\frac{1}{2% }}dydx=\int_{0}^{\frac{\pi }{2}}\int_{0}^{2\cos \theta }r\sin \theta \left( r\right) rdrd\theta =\int_{0}^{\frac{\pi }{2}}\int_{0}^{2\cos \theta }r^{3}\sin \theta drd\theta \end{equation*} \vspace{1pt} \begin{description} \item[{3 $\left[ 20\text{ pts.}\right] $}] Let $V$ be the volume in the first octant that is bounded by the cylinder $x^{2}+y^{2}=2y,$ the half-cone $z=% \sqrt{x^{2}+y^{2}}$ and \end{description} the $x,y-$plane. Give a triple integral expression in \textbf{cylindrical} coordinates for the volume of $V.$ Sketch $V.$ \textit{Do not evaluate the integral you give.} Solution: The equation of the cone is $z=r,$ and the equation of the circle is $r=2\sin \theta .$ Thus the region is as shown below. \ (Everything to the right of the x-z plane is shown, although the region requested is just the portion in the first octant.) $\left( r,\theta ,r\right) $\FRAME{dtbpFX}{4.4996in}{3in}{0pt}{}{}{Plot}{% \special{language "Scientific Word";type "MAPLEPLOT";width 4.4996in;height 3in;depth 0pt;display "USEDEF";plot_snapshots FALSE;mustRecompute FALSE;lastEngine "Maple";xmin "0";xmax "3.1416";ymin "0";ymax "3.1416";xviewmin "-3.26726399991353";xviewmax "3.26977727999827";yviewmin "-0.062855541071527E0";yviewmax "3.20568911079981";zviewmin "-0.062832";zviewmax "3.20568864";phi 66;theta 25;plottype 14;constrained TRUE;num-x-gridlines 25;num-y-gridlines 25;plotstyle "patch";axesstyle "normal";plotshading "XYZ";xis \TEXUX{z};yis \TEXUX{v952};var1name \TEXUX{$z$};var2name \TEXUX{$\theta $};function \TEXUX{$\left( r,\theta ,r\right) $};linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";coordinateSystem "cylindrical";var1range "0,3.1416";var2range "0,3.1416";surfaceColor "[linear:XYZ:RGB:0x00ff0000:0x000000ff]";surfaceStyle "Color Patch";num-x-gridlines 25;num-y-gridlines 25;surfaceMesh "Mesh";rangeset"XY";function \TEXUX{$\left( 2\sin \theta ,\theta ,z\right) $};linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";coordinateSystem "cylindrical";var1range "0,3.1416";var2range "0,5";surfaceColor "[linear:XYZ:RGB:0x00ff0000:0x000000ff]";surfaceStyle "Color Patch";num-x-gridlines 25;num-y-gridlines 25;surfaceMesh "Mesh";rangeset"X";}} Thus \begin{equation*} \text{volume}=\diiint\limits_{V}dV=\int_{0}^{\frac{\pi }{2}}\int_{0}^{2\sin \theta }\int_{0}^{r}rdzdrd\theta \end{equation*} \begin{description} \item[{4 [$15$ pts.]}] Give an integral expression in \textbf{spherical} coordinates for the volume of a sphere of radius $a.$ \textit{Do not evaluate this} \end{description} \textit{\ expression.} Solution: In spherical coordinates% \begin{equation*} \text{volume}=\diiint\limits_{V}\rho ^{2}\sin \phi dV \end{equation*}% so% \begin{equation*} \int_{0}^{2\pi }\int_{0}^{\pi }\int_{0}^{a}\rho ^{2}\sin \phi d\rho d\phi d\theta \end{equation*} \vspace{1pt} \end{document} %%%%%%%%%%%%% End /document/ma227_exam_2a_04s_v3_sol1.tex %%%%%%%%%%%%%