\hfill \thepage} %} \input{tcilatex} \begin{document} Ma227 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Final Exam \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ December 18, 2002 \vspace{1pt} NAME Pledge \vspace{1pt} DO ALL PROBLEMS. \ No Calculators Allowed. \ Each problem is worth 20 pts. Show all work. \vspace{1pt} I.) \ (3, 14, 3) \ Let $\overrightarrow{F}=(2xyz^{3}+e^{z})\overrightarrow{i}% +(x^{2}z^{3}+2y\ln z+3)\overrightarrow{j}$ \qquad \qquad \qquad \qquad \qquad \qquad $+(3x^{2}yz^{2}+xe^{z}+\frac{y^{2}% }{z}-2)\overrightarrow{k}$ \vspace{1pt} \qquad \qquad \qquad a.) \ Show that $\overrightarrow{F}$ is conservative. \qquad \qquad \qquad b.) \ Find a potential function for $\overrightarrow{F}% . $ \qquad \qquad \qquad c.) \ Find the work done in carrying a particle \qquad \qquad \qquad \qquad from $\left( 0,2,1\right) $ to $\left( 3,1,2\right) $ in this field. \pagebreak \vspace{1pt} II.) (10, 10) \ Set up, but do not evaluate, a double integral in Cartesian \qquad \qquad \qquad coordinates to find the area of the region in the first quadrant \qquad \qquad \qquad above $xy=1$ and $y=x;$ and below the line $y=2$ \qquad \qquad \qquad (i.) \ Integrating with respect to $^{\prime }x^{\prime }$ first. \qquad \qquad \qquad (ii) \ Integrating with respect to $^{\prime }y^{\prime }$ first. \qquad \qquad \qquad \lbrack\ i.e., Change the order of integration of part (i).\rbrack \pagebreak \vspace{1pt} III.) (10, 10) \ A lamina occupies the part of the disk $x^{2}+y^{2}\leq a^{2}$ \qquad \qquad \qquad\ \ that lies in the first quadrant. \ The density is $% xy^{2}.$ \vspace{1pt} \qquad \qquad \qquad a.) \ Find the mass of the lamina. \qquad \qquad \qquad b.) \ Set up, but do not evaluate, integrals in polar \qquad \qquad \qquad \qquad coordinates to find the center of mass of the lamina. \pagebreak \vspace{1pt} IV.) \ (8, 12) \ a.) \ Write the following triple integral in spherical coordinates. \qquad \qquad \qquad \qquad\ \ Do not evaluate. \vspace{1pt} \qquad \qquad \qquad \qquad \qquad $\iiint_{H}z^{3}\sqrt{x^{2}+y^{2}+z^{2}}% dV $ \vspace{1pt} \qquad \qquad \qquad \qquad where $H$ is the solid hemisphere with center at the origin and \qquad \qquad \qquad \qquad radius $^{\prime }1^{\prime }$ that lies above the $xy$-plane. \vspace{1pt} \qquad \qquad \qquad b.) \ Set up, but do not evaluate, a double integral in polar coordinates to \qquad \qquad \qquad \qquad find the surface area of the part of the sphere $% x^{2}+y^{2}+z^{2}=a^{2}$ \qquad \qquad \qquad \qquad that lies inside the cylinder $x^{2}+y^{2}=ax.$ \pagebreak V.) \ (20) \ Verify Green's Theorem for \vspace{1pt} \qquad \qquad \qquad \qquad $\int_{C}(x^{2}ydx-xy^{2}dy)$ \vspace{1pt} \qquad \qquad\ \ where $C$ is $x^{2}+y^{2}=4$, with counter-clockwise orientation. \pagebreak VI.) \ (5, 15) \ a.) \ State the Divergence Theorem. \qquad \qquad \qquad\ \ b.) \ Use the Divergence Theorem to evaluate $\int \int_{S}\overrightarrow{F}\cdot d\overrightarrow{S}$ \qquad \qquad \qquad \qquad\ \ where $\overrightarrow{F}=x^{3}% \overrightarrow{i}+2xz^{2}\overrightarrow{j}+3y^{2}z\overrightarrow{k}$ and $% S$ is the solid \qquad \qquad \qquad \qquad\ \ bounded by $z=4-x^{2}-y^{2}$ and the $xy$% -plane. \pagebreak VII.) \ (20) \ Use Stokes' Theorem to evaluate $\int_{C}\overrightarrow{F}% \cdot d\overrightarrow{r}$ where \qquad \qquad\ \ \ \ \ $\overrightarrow{F}=2z\overrightarrow{i}+4x% \overrightarrow{j}+5y\overrightarrow{k}$ and $C$ is the curve of intersection of \qquad \qquad\ \ \ \ \ $z=x+4$ and $x^{2}+y^{2}=4.$ \pagebreak VIII.) \ (20) \ Evaluate \ $\int \int xydS$ \ where $S$ is the boundary of the region enclosed \qquad \qquad\ \ \ \ \ \ by the cylinder $x^{2}+z^{2}=1$ and the planes $y=0$ and $x+y=2$. \qquad \qquad \qquad ($S$ consists of $3$ surfaces.) \vspace{1pt} \qquad \qquad \qquad You may use: \ \ $\int_{0}^{2\pi }\sin ^{2}\theta d\theta =\pi $ \vspace{1pt} \qquad \qquad \qquad \qquad \qquad \qquad \qquad $\int_{0}^{2\pi }\sin ^{3}\theta d\theta =0.$ \pagebreak IX.) (20)\ \ Solve: \vspace{1pt} \qquad \qquad \qquad \qquad $t\overrightarrow{x}$ $^{\prime }=\left( \begin{array}{ccc} 1 & 0 & 0 \\ 4 & 2 & -1 \\ 1 & 4 & 2 \end{array} \right) \overrightarrow{x}$ \vspace{1pt} \pagebreak X.) (20)\ Determine $e^{\overrightarrow{A}t}$ if $\overrightarrow{x}$ $% ^{\prime }=\overrightarrow{A}\overrightarrow{x}$ and $\overrightarrow{A}% =\left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right) .$ \end{document}