\documentclass{article}

\usepackage{graphics}

\renewcommand{\d}{\partial}
\renewcommand{\i}{\mathbf{i}}
\renewcommand{\j}{\mathbf{j}}
\renewcommand{\k}{\mathbf{k}}
\newcommand{\iint}{\int\!\!\!\int}
\newcommand{\iiint}{\int\!\!\!\int\!\!\!\int}

\setlength{\textheight}{9in}
\setlength{\voffset}{-.75in}
\setlength{\textwidth}{6.5in}
\setlength{\hoffset}{-.75in}

\def\problemspace{0.25in}

\begin{document}

%% TITLE

\noindent
\textbf{MATH 225: Calculus III}\hfill
Name:\makebox[2.5in][l]{\rule{2in}{.5pt}}

\medskip\noindent
\textbf{Exam II}\quad{\sl November 11, 2003}\hfill
Instructor:\makebox[2.5in][l]{\rule{2in}{.5pt}}

\vspace{.5in}
\noindent
Record your answers to the multiple choice problems by placing an $\times$
through one letter for each problem on this page. There are 8 multiple
choice questions worth 5 points each and 4 partial credits problems worth
10 points each.  You start with 20 points.  On the partial credit problems
try to simplify your answer and indicate your final answer clearly.
\textit{You must show your work and all important steps to receive credit}.

\medskip\noindent
You may use a calculator if you wish.

\vspace{.5in}

\def\answerspace{0.5in}


%% PROBLEM 1

Determine the equation of the line perpendicular to the surface $z =
1 + (x+y)e^y$ at the point $(0,0,1)$.

%% CORRECT

$x=t$, $y=t$, $z=1-t$

%% WRONG

$x=0$, $y=2t$, $z=1+t$

%% WRONG

$x=t$, $y=t$, $z=1$

%% WRONG

$x=t$, $y=2t$, $z=1-2t$

%% WRONG

$x=2t$, $y=t$, $z=1+2t$


%% PROBLEM 2

Find all the critical points of $f(x,y)=2y^3 + 3x^2y + 5y^2 + 2x^3$.

%% CORRECT

$(0,0)$, $(0,-\frac{5}{3})$, $(\frac{10}{9},-\frac{10}{9})$

%% WRONG

$(0,0)$

%% WRONG

$(0,0)$, $(0,-\frac{5}{3})$

%% WRONG

$(0,0)$, $(0,-\frac{5}{3})$, $(-\frac{10}{9},\frac{10}{9})$,
$(\frac{10}{9},-\frac{10}{9})$

%% WRONG

$(0,0)$, $(0,\pm\frac{5}{3})$, $(\pm\frac{10}{9},\pm\frac{10}{9})$


%% PROBLEM 3

Find the statement that describes the value of the iterated integral
$\int^2_0\int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}}2-\sqrt{4-x^2-y^2}\,dx\,dy$.

%% CORRECT

The volume of a half of a cylinder with a quarter of a sphere removed.

%% WRONG

The volume of a quarter of sphere.

%% WRONG

The volume of a cylinder with a half of a sphere removed.

%% WRONG

The volume of a half of a cylinder with an eighth of a sphere removed.

%% WRONG

The volume of an eighth of sphere.


%% PROBLEM 4

Reverse the order of integration in the iterated integral
$\int_0^2\int_{y^2-1}^{2y-1} f(x,y)\,dx\,dy$.

%% CORRECT
                                                                        
$\int_{-1}^3 \int^{\sqrt{x+1}}_{(x+1)/2} f(x,y)\,dy\,dx$

%% WRONG

$\int_{y^2-1}^{2y-1}\int_0^2  f(x,y)\,dy\,dx$.

%% WRONG

$\int_2^0\int_{2y-1}^{y^2-1}  f(y,x)\,dy\,dx$.

%% WRONG

$\int_{0}^4\int_{1+x/2}^{\sqrt{x+1}} f(x,y)\,dy\,dx$.

%% WRONG

$\int_{1}^3 \int_{1+\sqrt{x}}^{(x+1)/2} f(x,y)\,dy\,dx$


%% PROBLEM 5

Let $E$ be the solid in the first octant bounded by the cylinder $y^2 + z^2 =
4$ and the plane $y = 2x$.  Find the iterated integral that gives
$\iiint_E f(x,y,z)\,dV$.
\[ \resizebox{0.75in}{!}{\includegraphics{exam2.5.eps}} \]

%% CORRECT

$\int_0^1\int_{2x}^2\int_0^{\sqrt{4-y^2}}f(x,y,z)\,dz\,dy\,dx$

%% WRONG

$\int_0^2\int_0^{y/2}\int_x^{\sqrt{4-z^2}}f(x,y,z)\,dy\,dx\,dz$

%% WRONG

$\int_0^1\int_0^{2x}\int_0^{\sqrt{4-y^2}}f(x,y,z)\,dz\,dy\,dx$

%% WRONG

$\int_0^2\int_{y/2}^{1}\int_0^{\sqrt{4-z^2}}f(x,y,z)\,dy\,dx\,dz$

%% WRONG

$\int_0^1\int_{2x}^2\int_0^{2\sqrt{1-x^2}}f(x,y,z)\,dz\,dy\,dx$


%% PROBLEM 6

Determine which iterated integral gives the area of the part of the surface
$z = 1 + x^2 + y^2$ that lies above the square $0 \le x \le 1$, $0 \le y
\le 1$.
\[ \resizebox{0.75in}{!}{\includegraphics{exam2.6.eps}} \]

%% CORRECT

$\int_0^1\int_0^1\sqrt{1 + 4x^2 + 4y^2}\,dx\,dy$

%% WRONG

$\int_0^1\int_0^1\sqrt{1 + 2x + 2y}\,dx\,dy$

%% WRONG

$\int_0^1\int_0^1\sqrt{1 + x^2 + y^2}\,dx\,dy$

%% WRONG

$\int_0^1\int_0^1\int_0^{1+x^2+y^2}dz\,dx\,dy$

%% WRONG

$\int_0^1\int_0^1\int_1^{3}2\sqrt{x^2+y^2}\,dz\,dx\,dy$


%% PROBLEM 7

Determine which iterated integrals gives the mass of the solid region
inside the sphere $x^2 + y^2 + z^2=4$ and above the cone $3z^2=x^2+y^2$
if the density is $\delta(x,y,z)=x^2z$.

\[ \resizebox{0.75in}{!}{\includegraphics{exam2.7.eps}} \]

%% CORRECT

$\int_0^{2\pi} \int_0^{\pi/3} \int_0^2
\rho^5 \sin^3(\phi)\cos(\phi)\cos^2(\theta)\,d\rho\,d\phi\,d\theta$

%% WRONG

$\int_0^{2\pi} \int_0^{\pi/4} \int_0^2
\rho^3 \sin^2(\phi)\cos(\phi)\cos^2(\theta)\,d\rho\,d\phi\,d\theta$

%% WRONG

$\int_0^{2\pi} \int_0^{\pi/3} \int_{\sqrt{3}}^2
\rho^5 \cos^3(\phi)\sin(\phi)\cos^2(\theta)\,d\rho\,d\phi\,d\theta$

%% WRONG

$\int_0^{2\pi} \int_0^{\pi/6} \int_{\sqrt{3}}^2
\rho^3 \cos^2(\phi)\sin(\phi)\sin^2(\theta)\,d\rho\,d\phi\,d\theta$

%% WRONG

$\int_0^{2\pi} \int_0^{\pi/6} \int_0^2
\rho^5 \sin^2(\phi)\cos(\phi)\cos^2(\theta)\,d\rho\,d\phi\,d\theta$


%% PROBLEM  8

Calculate the volume of the solid region between the surfaces $z=6-x^2-y^2$
and $z=\sqrt{x^2+y^2}$ over the domain $1 \le x^2 + y^2 \le 4$.

%% CORRECT

$35\pi/6$

%% WRONG

$13\pi/3$

%% WRONG

$3\pi$

%% WRONG

$8\pi/3$

%% WRONG

$71\pi/6$


%% END


\newpage
%\bigskip

\noindent\llap{9.\ }Find the local maximum and minimum values and saddle
points, if any, of the function $f(x,y)=x^2 + 2y^2 - x^2y$.

\newpage
%\bigskip

\noindent\llap{10.\ }Determine the minimum distance to the origin of the
points on the curve defined by $x^2 - 7xy + y^2 = 36$.

\newpage
%\bigskip

\noindent\llap{11.\ }Let $D$ be the region between the circles $x^2 + y^2 =
4$ and $x^2 + y^2 = 2x$ in the first quadrant. The area of $D$ is $\pi/2$.
Find the average value of $f(x,y) = y$ over $D$.
\[ \resizebox{1.5in}{!}{\includegraphics{exam2.11.eps}} \]

\newpage
%\bigskip

\noindent\llap{12.\ }
Compute the double integral $\iint_D (x+y)\,dA$ where $D$
is the region bounded by the ellipse $(x-2)^2+4(y-1)^2=16$ by
making the change of variables $x=2+4r\cos \theta$, $y=1+2r\sin
\theta.$

\end{document}
