\input calc
\input epsf
\def\problemspace{0.75truein}

%% TITLE
{\bf Math 225: Calculus III}\hfill Name:\hbox to 2.5truein{\hrulefill}\medskip
{\bf Exam III}\quad{\sl April 20, 1995}\hfill Section:\hbox to 2.5truein{\hrulefill}

\bigskip\bigskip
Record your answers to the multiple choice problems by placing an
$\times$ through one letter for each problem on this answer sheet.
There are 15 multiple choice questions worth 6 points each. You
start with 10 points.

\bigskip\bigskip

\def\answerspace{0.25in}


%1
%% PROBLEM
Evaluate $\int_0^1\int_{-x}^{2x}\int_0^{x+y+1}2(z-1)\,dz\,dy\,dx$.
%% CORRECT
{${3\over4}$}
%% WRONG
{${5\over6}$}
%% WRONG
{${7\over8}$}
%% WRONG
{${15\over16}$}
%% WRONG
{${23\over24}$}

%2
%% PROBLEM
Let $D$ be the solid bounded by the planes $x+y+z = 4$, 
$y=x$, $x=0$, and $z=0$. Find the integral that gives
the volume of $D$.
%% CORRECT
{$\int_0^2\int_x^{4-x}\int_0^{4-x-y}1\,dz\,dy\,dx$}
%% WRONG
{$\int_0^4\int_0^y\int_0^{2-x-y}1\,dz\,dx\,dy$}
%% WRONG
{$\int_0^4\int_0^y\int_0^{x+y+z}1\,dz\,dx\,dy$}
%% WRONG
{$\int_0^2\int_x^2\int_0^{2-x}1\,dz\,dy\,dx$}
%% WRONG
{$\int_0^2\int_0^{y-x}\int_0^{2-x-y}1\,dz\,dx\,dy$}

%3
%% PROBLEM
Let $D$ be the solid box bounded by $z = 3$, $x=1$, and $y=1$ in the first octant.
Suppose the density function of $D$ is $\delta(x,y,z) = 2z$. Compute the
center of gravity of~$D$.
%% CORRECT 
{$({1\over2},{1\over2},{2})$}
%% WRONG
{$({1\over2},{1\over2},{1})$}
%% WRONG
{$({1\over2},{1\over2},{3\over2})$}
%% WRONG
{$({1\over2},{1\over2},{1\over2})$}
%% WRONG
{$({1\over2},{1\over2},{5\over2})$}


%4
%% PROBLEM
Let $D$ be the portion of the solid sphere of radius 3 that lies inside the
cylinder $x^2 + y^2 = 2$. Determine which of the following integrals gives
$\iiint_D x y z\,dV$.

\centerline{\epsfxsize=1.5in\epsfbox{exam3.4.eps}}

%% CORRECT
{$\int_0^{2\pi}\int_0^{\sqrt{2}}\int_{-\sqrt{9-r^2}}^{\sqrt{9-r^2}}
r^3\cos(\theta)\sin(\theta)z\,dz\,dr\,d\theta$}
%% WRONG
{$\int_0^{2\pi}\int_0^{2}\int_{-\sqrt{3-r^2}}^{\sqrt{3-r^2}}
r^2\cos(\theta)\sin(\theta) \,dz\,dr\,d\theta$}
%% WRONG
{$\int_0^{2\pi}\int_0^{\sqrt{2}}\int_{-\sqrt{9-r^2\cos^2(\theta)}}^{\sqrt{9-r^2\cos^2(\theta)}}
r^2z\,dz\,dr\,d\theta$}
%% WRONG
{$\int_0^{\pi}\int_{-\sqrt{2}}^{\sqrt{2}}\int_{-\sqrt{3-r\cos(\theta)}}^{\sqrt{3-r\cos(\theta)}}
r^3\cos(\theta)\sin(\theta)z\,dz\,dr\,d\theta$}
%% WRONG
{$\int_0^{2\pi}\int_{-2}^{2}\int_{-3}^{3}
r^2\cos(\theta)\sin(\theta)\,dz\,dr\,d\theta$}

%5
%% PROBLEM
Convert the integral $\int_{-3}^3\int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}}
\int_{\sqrt{3(x^2+y^2)}}^{\sqrt{36 - x^2-y^2}}(x^2 + y^2 + z^2)^{3/2}\,dz\,dy\,dx$
to spherical coordinates.

\centerline{\epsfxsize=1.5in\epsfbox{exam3.1.eps}}

%% CORRECT
{$\int_0^{2\pi}\int_0^{\pi/6}\int_0^6
\rho^5\sin(\phi)\,d\rho\,d\phi\,d\theta$}
%% WRONG
{$\int_0^{2\pi}\int_0^{\pi/3}\int_0^6
\rho^2\sin(\phi)\,d\rho\,d\phi\,d\theta$}
%% WRONG
{$\int_0^{\pi}\int_0^{\pi}\int_0^6
\rho^{3/2}\sin(\phi)\,d\rho\,d\phi\,d\theta$}
%% WRONG
{$\int_0^{2\pi}\int_0^{\pi/2}\int_0^{36}
\rho^3\sin(\phi)\,d\rho\,d\phi\,d\theta$}
%% WRONG
{$\int_0^{2\pi}\int_0^{\pi/4}\int_0^{36}
\rho^4\sin(\phi)\,d\rho\,d\phi\,d\theta$}



%6
%% PROBLEM
Consider the change of variables $u = e^{x+y}$, $v = e^{x-y}$.
Compute the Jacobian determinate ${\partial(x,y)\over \partial(u,v)}$.
%% CORRECT
{$-{1\over 2uv}$}
%% WRONG
{$e^{2x}+e^{-2y}$}
%% WRONG
{$0$}
%% WRONG
{${1\over2}({1\over u} + {1\over v})$}
%% WRONG
{$e^{x+y} - e^{x-y}$}

%7
%% PROBLEM
Identify which of the following plots represents the vector field
$\F(x,y) = {x\over\sqrt{x^2+y^2}}\i - {y\over\sqrt{x^2+y^2}}\j$.
%% CORRECT
\vtop{\epsfxsize=1.5in\epsfbox{exam3.2.1.eps}}
%% WRONG
\vtop{\epsfxsize=1.5in\epsfbox{exam3.2.2.eps}}
%% WRONG
\vtop{\epsfxsize=1.5in\epsfbox{exam3.2.3.eps}}
%% WRONG
\vtop{\epsfxsize=1.5in\epsfbox{exam3.2.4.eps}}
%% WRONG
\vtop{\epsfxsize=1.5in\epsfbox{exam3.2.5.eps}}

%8
%% PROBLEM
Let $\F(x,y,z) = x^3y\i + (y-z)\j + k$. Compute $\curl\F$.
%% CORRECT
{$\i-x^3\k$}
%% WRONG
{$3x^2y\i + \j$}
%% WRONG
{$-\i + 3x^2y\j + (x^3y - 3x^2y(y-z))\k$}
%% WRONG
{$\0$}
%% WRONG
{$x^3\i - \j$}


%9
%% PROBLEM
Let $\F(x,y,z) = \cos(xy)\i + \sin(yz)\j + (x+z)y\k$. Compute $\div\F$.
%% CORRECT
{$y(1-\sin(xy))+z\cos(yz)$}
%% WRONG
{$-y\sin(xy)\i+z\cos(yz)\j + y\k$}
%% WRONG
{$(x+z-y\cos(yz))\i - y\j + x\sin(xy)\k$}
%% WRONG
{$0$}
%% WRONG
{$-\sin(xy) + \cos(xy) + y$}


%10
%% PROBLEM
Let $\CC$ be the curve parameterized by $\r(t) = (t^3-t^2)\i + t^2\j$, $-1\le t \le 1$.
Determine which of the following integrals gives the line integral
$\int_\CC x + y \, ds$.
%% CORRECT
{$\int_{-1}^1 t^4\sqrt{9t^2-12t+8}\,dt$}
%% WRONG
{$\int_{-1}^1 t^3\,dt$}
%% WRONG
{$\int_{-1}^1 3t^2\,dt$}
%% WRONG
{$\int_{-1}^1 t^3\sqrt{t^6 - 2t^5 + t^4}\,dt$}
%% WRONG
{$\int_{-1}^1t^5(9t^2-12t +8)\,dt$}

%11
%% PROBLEM
Let $\CC$ be the curve parameterized by $\r(t) = e^{t^2}\i + e^{-t^2}\j + t\k$,
$0\le t \le1$. Use the Fundamental Theorem of Line Integrals to  calculate the integral
$\int_\CC y(z^2+1)\,dx + x(z^2+1)\,dy + 2z(xy+1)\,dz$.
%% CORRECT
{$2$}
%% WRONG
{$1$}
%% WRONG
{$0$}
%% WRONG
{$1-e^{-1}$}
%% WRONG
{$e+e^{-1}+1$}


%12
%% PROBLEM
Let $\CC$ be the counterclockwise path around the perimeter of the
triangle with vertices $(0,0)$, $(1,0)$, and $(1,2)$. Use Green's Theorem to calculate
$\int_\CC (y + x\sin(y^2))\,dx + x^2y\cos(y^2)\,dy$.

\centerline{\epsfxsize=1in\epsfbox{exam3.3.eps}}

%% CORRECT
{$-1$}
%% WRONG
{$0$}
%% WRONG
{$-2$}
%% WRONG
{$\sin(4)-2\cos(4)$}
%% WRONG
{$\cos(4) - \sin(4) - 1$}

%13
%% PROBLEM
Let $\CC$ be the curve parameterized by $\r(t) = t^2\i + t\j$, $0\le t \le 3$,
and let $\F(x,y) = (x-y)\i + (x+y)\j$. Compute the flow integral $\int_\CC \F\cdot d\r$.
%% CORRECT
{36}
%% WRONG
{24}
%% WRONG
{12}
%% WRONG
{9}
%% WRONG
{0}


%14
%% PROBLEM
Let $f(x,y,z)$ be a function and let $\F(x,y,z)$ be a vector field.
Determine which of the following expressions is {\bf not} defined.
%% CORRECT
{$\curl\div\F$}
%% WRONG
{$\div\curl\F$}
%% WRONG
{$\div\grad f$}
%% WRONG
{$\curl\grad f$}
%% WRONG
{$\grad\div\F$}

%15
%% PROBLEM
Which of the following gives the change from spherical coordinates $(\rho,\phi,\theta)$
to rectangular coordinates $(x,y,z)$.
%% CORRECT
\vtop{\hsize = 1.5truein$\eqalign{x &= \rho\sin(\phi)\cos(\theta)\cr
y &= \rho\sin(\phi)\sin(\theta)\cr z &= \rho\cos(\phi)}$}
%% WRONG
\vtop{\hsize = 1.5truein$\eqalign{x &= \rho\cos(\phi)\cos(\theta)\cr
y &= \rho\cos(\phi)\sin(\theta)\cr z &= \rho\sin(\phi)}$}
%% WRONG
\vtop{\hsize = 1.5truein$\eqalign{x &= \rho\cos(\phi)\cr
y &= \rho\sin(\phi)\cr z &= \sqrt{\rho^2 - 1}}$}
%% WRONG
\vtop{\hsize = 1.5truein$\eqalign{x &= \rho\cos(\phi)\sin(\theta)\cr
y &= \rho\cos(\phi)\cos(\theta)\cr z &= \rho^2\sin(\phi)}$}
%% WRONG
\vtop{\hsize = 1.5truein$\eqalign{x &= \rho\cos(\phi)\cos(\theta)\cr
y &= \rho\sin(\phi)\sin(\theta)\cr z &= \rho\tan(\phi)}$}


%% END
\bye
