\magnification=\magstep1
\input epsf
\input calc
\def\overd{\over \displaystyle}
\def\problemspace{0.2in}

%% TITLE
\noindent
{\bf Math 225: Calculus III}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam III}\quad{\sl April 17, 1997}\hfill
Section:\hbox to 2.5truein{\hrulefill}

\bigskip\noindent
Record your answers to the multiple choice problems by placing an
$\times$ through one letter for each problem on this answer sheet.
There are 14 multiple choice questions worth 5 points each. You start
with 30 points.

\def\answerspace{0.25in}


%% PROBLEM
Calculate the area bounded by the spirals $r = \theta$ and $r = 2\theta$
and the positive $x$-axis.

$$\epsfxsize=3truein\epsfbox{exam3.1.eps}$$

%% CORRECT
$4\pi^3$
%% WRONG
$\pi^2/2$
%% WRONG
$2\pi^2$
%% WRONG
$\pi^3/2$
%% WRONG
$\pi^2$

%% PROBLEM
Let $D$ be the solid region bounded by the planes $x + 2y + z = 3$,
$2x + y - z = 3$, $x = 0$, and $y = 0$. Determine which of the
following integrals gives $\iiint_D f(x,y,z)\,dV$.

$$\epsfxsize=1.5truein\epsfbox{exam3.2.eps}$$

%% CORRECT
$\int_0^2\!\!\int_{0}^{2-x}\!\!\int_{2x+y-3}^{3-x-2y}\!\!f(x,y,z)dzdydx$
%% WRONG
$\int_0^x\!\!\int_{0}^{y}\!\!\int_{2x+y+z}^{x+2y+z}\!\!f(x,y,z)dzdydx$
%% WRONG
$\int_0^3\!\!\int_{0}^{3-2x}\!\!\int_{0}^{3-x-2y}\!\!f(x,y,z)dzdydx$
%% WRONG
$\int_0^{3/2}\!\!\int_{0}^{(3-x)/2}\!\!\int_{0}^{3-x-2y}\!\!f(x,y,z)dzdydx$
%% WRONG
$\int_0^3\!\!\int_{3-2x}^{(3-x)/2}\!\!\int_{2x+y-3}^{3-x-2y}\!\!f(x,y,z)dzdydx$

%% PROBLEM
Evaluate the iterated integral $\int_0^1\int_0^z\int_0^y
e^{x-y}dx\,dy\,dz$.

%% CORRECT
${1\over2} - e^{-1}$
%% WRONG
$1 - e^{-1}$
%% WRONG
$e-{1\over2}$
%% WRONG
${3\over2} + e^{-1}$
%% WRONG
$e^{-1}-1$

%% PROBLEM
Consider the region defined by the inequalities
$2x - 1 \le y \le  2x$ and $-x^2 \le y \le 4 - x^2$
$$\epsfxsize=1truein\epsfbox{exam3.4.eps}$$
Using the change of variables
$$\vbox{\halign{\hfil$#$ & $#$\hfil &\qquad \hfil$#$ & $#$\hfil\cr 
u &= 2x - y  & x &= -1 + \sqrt{1 + u + v}\cr
v &= x^2 + y & y &= \sqrt{1 + u + v} - u - 2\cr
}}$$
transform the integral $\iint_R (x+1)^2\,dA$ into an iterated integral in
$u$ and $v$.

%% CORRECT
{${1\over2}\int_0^4\!\int_0^1 \sqrt{1+u+v}\,du\,dv$}
%% WRONG
{$\int_0^{4}\!\int_0^1 \sqrt{1+u+v}\,du\,dv$}
%% WRONG
{${1\over2}\int_0^1\!\int_0^4 1+u+v\,du\,dv$}
%% WRONG
{$\int_0^1\!\int_0^4 \sqrt{1+u+v}\,du\,dv$}
%% WRONG
{$\int_0^{4}\!\int_0^1 1+u+v\,du\,dv$}


%% PROBLEM
Suppose the density of the solid unit sphere $x^2+y^2+z^2\le1$ is given by
$\delta(x,y,z) = 1 + z$. Find its center of mass.

%% CORRECT
$(0,0,{1\over5})$
%% WRONG
$(0,0,0)$
%% WRONG
$(0,0,{1\over 10\pi})$
%% WRONG
$(0,0,{2\over 15})$
%% WRONG
$(0,0,{3\over 4\pi})$


%% PROBLEM
Find the vector field that is represented in the following plot.

$$\epsfxsize=2truein\epsfbox{exam3.5.eps}$$

%% CORRECT
${x\over\sqrt{x^2+y^2}}\i - {y\over \sqrt{x^2+y^2}}\j$
%% WRONG
${x\over\sqrt{x^2+y^2}}\i + {y\over \sqrt{x^2+y^2}}\j$
%% WRONG
${y\over\sqrt{x^2+y^2}}\i + {x\over \sqrt{x^2+y^2}}\j$
%% WRONG
${y\over\sqrt{x^2+y^2}}\i - {x\over \sqrt{x^2+y^2}}\j$
%% WRONG
${x+y\over\sqrt{2(x^2+y^2)}}\i - {x-y\over\sqrt{2(x^2+y^2)}}\j$

%% PROBLEM
Determine which of the following sets of inequalities describes
the solid region bounded above by the sphere $x^2 + y^2 + z^2 = 8$
and below by the paraboloid $2z = x^2 + y^2$.

%% CORRECT
\enspace\hbox to1.5truein{\vtop{\halign{\hfil$#$\hfil&$#$\hfil\cr
{1\over 2}r^2 \le &z \le \sqrt{8 - r^2}\cr
0 \le &r \le 2\cr
0 \le &\theta \le 2\pi\cr}}\hfil}

%% WRONG
\enspace\hbox to1.5truein{\vtop{\halign{\hfil$#$\hfil&$#$\hfil\cr
0 \le &\rho \le 2\sqrt{2}\cr
0 \le &\phi \le \pi/4\cr
0 \le &\theta \le 2\pi\cr}}\hfil}

%% WRONG
\enspace\hbox to1.5truein{\vtop{\halign{\hfil$#$\hfil&$#$\hfil\cr
0 \le &z \le \sqrt{8 - r^2}\cr
0 \le &r \le 4\cr
0 \le &\theta \le \pi\cr}}\hfil}

%% WRONG
\enspace\hbox to1.5truein{\vtop{\halign{\hfil$#$\hfil&$#$\hfil\cr
{1\over 2}r \le &\rho \le 2\sqrt{2}\cr
0 \le &\phi \le \pi/2\cr
0 \le &\theta \le \pi\cr}}\hfil}

%% WRONG
\enspace\hbox to1.5truein{\vtop{\halign{\hfil$#$\hfil&$#$\hfil\cr
{1\over 2}r^2 \le &z \le 2\sqrt{2}\cr
0 \le &r \le \sqrt{2}\cr
0 \le &\theta \le 2\pi\cr}}\hfil}

%% PROBLEM
Calculate the volume of the solid region in the first octant that lies
inside the cylinder, $x^2+y^2=1$, and below the plane, $x+y+z=2$.

$$\epsfxsize=2truein\epsfbox{exam3.3.eps}$$

%% CORRECT
${\pi\over 2} - {2\over 3}$
%% WRONG
${\pi} - {1\over 3}$
%% WRONG
$2\pi - {4\over 3}$
%% WRONG
${\pi-2\over 3}$
%% WRONG
${\pi-1\over 2}$

%% PROBLEM
Let $\F(x,y,z) = xy^3 \i + (x^2 + z^2)\j + x^3 z \k$. Compute $\div \F$.

%% CORRECT
$x^3 + y^3$
%% WRONG
$-2z\i -3x^2z\j + (2x-3xy^2)\k$
%% WRONG
$(3x^2y+y^3)\i + 2(x+z)\j + (x^3 + 3x^2z)\k$
%% WRONG
$x^3+y^3 + 3xy^2 + 3x^2z + 2x+2z$
%% WRONG
$-2\i - 3x(x+2z)\j + (2 - 6 xy - 3y^2)\k$


%% PROBLEM
Let $\F = (x^2+y^3z)\i + (xz-y)\j + x^2yz\k$. Compute $\curl\F$.

%% CORRECT
$x(xz-1)\i + y(y^2 - 2 x z)\j + z(1 - 3y^2)\k$
%% WRONG
$x^2y + 2x - 1$
%% WRONG
$(y^3 + 3y^2z + 2x)\i + (x+z-1)\j+x(x y + xz + 2yz)\k$
%% WRONG
$2x\i - \j + x^2y)\k$
%% WRONG
${\bf 0}$


%% PROBLEM
Let $\CC$ be the directed line segment from the point $(-1,0,4)$ to
the point $(3,2,0)$. Compute the value of line integral $\int_\CC (x-2y)z\,ds$.

%% CORRECT
$-12$
%% WRONG
$-6$
%% WRONG
$1/2$
%% WRONG
$1/3$
%% WRONG
$1$

%% PROBLEM
Compute the flow integral $\int_\CC\F\cdot d\r$ of the velocity field
$\F(x,y,z) = z\i + x\j + y\k$ along the curve $\r(t) =  t\i + t^2\j +
t^3\k$, $-1 \le t \le 1$.

%% CORRECT
$2.53$
%% WRONG
$1.42$
%% WRONG
$3.11$
%% WRONG
$0.87$
%% WRONG
$2.00$

%% PROBLEM
Use the Fundamental Theorem of Line Integrals to evaluate
$$\int_\CC (2xy+z)\,dx + (x^2+3y^2+z)\,dy + (x+y-1)\,dz$$
where $\CC$ is the curve $\r(t) = 3te^{(t-1)^3}\i + 2te^{(t-1)^2}\j +
te^{(t-1)}\k$, $0 \le t \le 1$.

%% CORRECT
$30$
%% WRONG
$12$
%% WRONG
$10$
%% WRONG
$6$
%% WRONG
$5$

%% PROBLEM
Let $R$ be the rectangular region defined by $-1\le x\le2$, $-2\le
y\le3$ and let $\CC$ be the boundary of $R$ oriented counterclockwise.
Use Green's Theorem to compute $\int_\CC x y\,dx - (x^2+y^2)\,dy$

%% CORRECT
$-22.5$
%% WRONG
$-18.0$
%% WRONG
$15.5$
%% WRONG
$29.0$
%% WRONG
$-14.5$

%% END
\end
