\hfill \thepage} %} \input{tcilatex} \begin{document} Ma227\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad Test \ 1\qquad \qquad \qquad\ July 28, 2003 \vspace{1pt} Name: Pledge: \vspace{1pt} SHOW ALL WORK. \ \ NO CALCULATORS ALLOWED. \vspace{1pt} I.) \ \ (20) \qquad Reverse the order of integration and evaluate: \vspace{1pt} \qquad \qquad \qquad \qquad $\int_{0}^{1}\int_{x}^{1}e^{\frac{x}{y}}dydx$ \vspace{1pt} \pagebreak \vspace{1pt} II.) \ \ (20) \qquad Find, by double integration, the area outside \ $x^{2}+y^{2}=1$ \qquad and inside \ \ $r=1+\sin \theta .$ \vspace{1pt} \pagebreak \vspace{1pt} III.) \ \ (12, 12, 12) \vspace{1pt} \qquad \qquad a.) \ Set up a triple integral in cylindrical coordinates to find the \qquad \qquad \qquad volume of the region bounded by \ \ $z=x^{2}+y^{2}$ \ and \qquad \qquad \qquad $z=36-3x^{2}-3y^{2}.$ \ \ \ DO\ NOT EVALUATE. \vspace{1pt} \qquad \qquad b.) \ Set up a triple integral in spherical coordinates to find the \qquad \qquad \qquad volume of the region above \ \ $z^{2}=x^{2}+y^{2}$ \ \ and below \qquad \qquad \qquad $x^{2}+y^{2}+z^{2}=18$ \ in the first octant. \ \ DO\ NOT\ EVALUATE. \vspace{1pt} \qquad \qquad c.) \ Set up a double integral to find the surface area of \ \ $z=x^{2}+y$ \qquad \qquad \qquad that lies above the triangle with vertices \ $\left( 0,0\right) ,$ $\left( 1,0\right) ,$ and \ $\left( 0,2\right) .$ \qquad \qquad \qquad DO\ NOT\ EVALUATE. \pagebreak \pagebreak \vspace{1pt} IV.) \ \ (10, 14) \vspace{1pt} \qquad \qquad a.) \ Write in spherical coordinates \ \ (Solve for $\rho $ \ in terms of \ $\phi $) \vspace{1pt} \qquad \qquad \qquad \qquad $x^{2}+y^{2}=9z+9$ \vspace{1pt} \qquad \qquad b.) \ Change to polar coordinates: \ (DO\ NOT\ EVALUATE) \vspace{1pt} \qquad \qquad \qquad \qquad $\int_{0}^{8}\int_{\frac{y^{2}}{4}}^{2y}dxdy$ \end{document}