\nopagenumbers
\def\pt#1{\hbox{#1) }}
\input amssym.def
\input amssym.tex
\def\R{{\Bbb R}}
\font\srm=cmr8
\magnification=\magstep1

\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 37\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 37} 
\bigskip


\item{1.}  Sketch the regions that give rise to the following
integrals.  Then rewrite it as a double integral with the integration
order reversed.
$$
\eqalign{
\pt a &\int_{0}^{1}\int_{\sqrt x}^{x^2}f(x,y)\,dy\,dx\cr\cr
\pt b &\int_{0}^{2\pi}\int_{0}^{|\cos x|}f(x,y)\,dy\,dx\cr\cr
\pt c &\int_{a}^{b}\int_{-a}^{-b}f(x,y)\,dx\,dy\cr}\qquad
\eqalign{
\pt d &\int_{-1/\sqrt2}^{1/\sqrt2}\int_{\sqrt{1-x^2}}^{\sqrt{4-x^2}}
            f(x,y)\,dy\,dx\cr\cr
\pt e &\int_{2-\sqrt3}^{2+\sqrt3}\int_{1/y}^{4-y}
       f(x,y)\,dx\,dy\cr\cr
\pt f &\int_{1}^{2}\int_{\ln y}^{e^y}f(x,y)\,dx\,dy\cr}\hskip1.1in
$$

\medskip
\item{2.}Evaluate:
$$
\eqalign{
\pt a &\int_{0}^{\pi}\int_{y}^{\pi}{\sin x\over x}\,dx\,dy\cr\cr
\pt c &\int_{0}^{\ln 16}\int_{e^{x/2}}^{4}{dy\,dx\over\ln y}\cr}\hskip1in
\eqalign{
\pt b &\int_{0}^{2}\int_{x}^{2}e^{-y^2/2}\,dy\,dx\cr\cr
\pt d &\int_{0}^{1}\int_{y^2}^{1}{e^x\over\sqrt x}\,dx\,dy\cr}\hskip1.36in
$$

\medskip
\itemitem{3.\hskip12pt a)}  Sketch the set of all points $(x,y)$ in
Euclidean coordinates on the plane that satisfy the inequalities
$$
x_0\leq x\leq x_0+\Delta x
\qquad\hbox{and}\qquad 
y_0\leq y\leq y_0+\Delta y.
$$

\itemitem{b)}  Sketch the set of all points $(r,\theta)$ in
polar coordinates on the plane that satisfy the inequalities
$$
r_0\leq r\leq r_0+\Delta r
\qquad\hbox{and}\qquad 
\theta_0\leq \theta\leq \theta_0+\Delta \theta.
$$

\itemitem{c)}  Find the area of the ring between two circles, one of
radius $r_0$, the other of radius $r_0+\Delta r$.

\medskip
\itemitem{d)}  What fraction of the area in c) is included between two
rays whose angles differ by $\Delta\theta$?

\medskip
\itemitem{e)}  Show that the area of the region in part b) is
precisely
$$
\left(r_0+{\Delta r\over2}\right)\Delta r\Delta\theta.
$$

\medskip
\item{4.}  Sketch the regions that give rise to the following
integrals:
$$
\pt a \int_{0}^{\pi}\int_{0}^{2a\sin\theta}r\,dr\,d\theta\hskip.5in
\pt b \int_{-\pi/4}^{\pi/4}\int_{1}^{2}r\,dr\,d\theta\hskip.5in
\pt c \int_{0}^{\pi/4}\int_{0}^{b\sec\theta}r\,dr\,d\theta\hskip.2in
$$

\medskip
\item{5.}  Carry out the integrals in Problem 4.

\bigskip
\itemitem{6.\hskip12pt a)}  Find the volume of the solid in the first
octant bounded by the cylinder $x^2+y^2=4$ and the parabaloid
$z=x^2+y^2$.

\medskip
\itemitem{b)}  Find the volume of the solid bounded above by
$z=1-x^2-y^2$ and below by the $xy$-plane

\bigskip
\item{7.} Integrate the function $f$ over the region $\Omega$ by
changing to polar coordinates:
$$
\eqalign{
\pt a f(&x,y)={1\over4a}(x^2+y^2)\cr
      \Omega&=\hbox{ disk of radius $a$ centered at $(0,a)$}\cr
\pt b f(&x,y)=y\cr
      \Omega&=\hbox{ region between the circles of radii $a$ and $b$
                     centered at the origin ($a