
\input calc
\let\ds=\displaystyle
\def\problemspace{.25in}

%% TITLE
{\bf Math 225: Calculus III}\hfill Name:\hbox to 2.5truein{\hrulefill}\medskip
{\bf Exam III}\quad{\sl December 6, 1990}\hfill Score:\hbox to 2.5truein{\hrulefill}

\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 16 multiple
choice questions worth 6 points each. You start with 6 free points.

%1
%% PROBLEM
Evaluate $\iint_R x\,dA$ where $R$ is the semi-annular region
in the first quadrant between the circles of radius 1 and
radius 3.
%% CORRECT
{$26\over\ds3$}
%% WRONG
{$31\over\ds3$}
%% WRONG
{$9$}
%% WRONG
{$25\over\ds3$}
%% WRONG
{$10$}


%2
%% PROBLEM
Determine which of the following integrals gives the surface area of the portion of the graph of $f(x,y) = e^x\cos(y)$ lying over a region~$R$.
%% CORRECT
{$\iint_R\sqrt{1+e^{2x}}\,dA$}
%% WRONG
{$\iint_R\sqrt{1+e^x\cos(y) - e^x\sin(y)}\,dA$}
%% WRONG
{$\iint_R e^x\cos(y)\,dA$}
%% WRONG
{$\iint_R e^x\,dA$}
%% WRONG
{$\iint_R 1+e^x\,dA$}

%3
%% PROBLEM
Let $D$ be the solid bounded by the planes
$x+y+z = 2$, $y = 0$, $y = x$, and $z = 0$.
Find the integral that gives the volume of~$D$.
%% CORRECT
{$\int_0^1\int_y^{2-y}\int_0^{2-x-y}1\,dz\,dx\,dy$}
%% WRONG
{$\int_0^2\int_0^x\int_0^{2-x-y}1\,dz\,dy\,dx$}
%% WRONG
{$\int_0^2\int_0^x\int_0^{x+y+z}1\,dz\,dy\,dx$}
%% WRONG
{$\int_0^1\int_y^2\int_0^{2-y}1\,dz\,dx\,dy$}
%% WRONG
{$\int_0^1\int_0^{x-y}\int_0^{2-x-y}1\,dz\,dy\,dx$}

%4
%% PROBLEM
Let $D$ be the portion of the ball $x^2 + y^2 + z^2 \le 1$
cut out by the cylinder $x^2 + y^2 = x$.
Write $\iiint_D xyz\,dV$ as an iterated integral in
cylindrical coordinates.
%% CORRECT
{$\int_0^\pi\int_0^{\cos(\theta)}\int_{-\sqrt{1-r^2}}^{\sqrt{1-r^2}}
zr^3\cos(\theta)\sin(\theta)\,dz\,dr\,d\theta$}
%% WRONG
{$\int_0^{2\pi}\int_0^1\int_{-\sqrt{1-r^2}}^{\sqrt{1-r^2}}
zr^2\cos(\theta)\sin(\theta)\,dz\,dr\,d\theta$}
%% WRONG
{$\int_0^{2\pi}\int_0^{\cos(\theta)}\int_0^{\sqrt{1-r^2}}
zr^2\cos(\theta)\sin(\theta)\,dz\,dr\,d\theta$}
%% WRONG
{$\int_0^\pi\int_0^{r\cos(\theta)}\int_0^{\sqrt{1-r^2}}
zr^3\cos(\theta)\sin(\theta)\,dz\,dr\,d\theta$}
%% WRONG
{$\int_0^{2\pi}\int_0^1\int_0^{\sqrt{1-r^2}}
r\cos(\theta)\sin(\theta)\,dz\,dr\,d\theta$}


%5
%% PROBLEM
Let $D$ be the solid region bounded above by the sphere $x^2 + y^2 + z^2 = 36$
and below by the upper nappe of the cone $3(x^2 + y^2) = z^2$. Write
$\iiint_D(x^2 + y^2 + z^2)^{3/2}\,dV$ as an iterated integral in spherical
coordinates.
%% 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
Let $D$ be the portion of the solid cylinder $x^2 + y^2 \le 1$ below $z
= 2$ in the first octant. Suppose the density at a point in $D$ is twice
the distance of the point to the $z$-axis. Compute the center of gravity
of~$D$, assuming its total mass is~$2\pi\over\ds3$. (Use symmetry.)
%% CORRECT
{$({3\over 2\pi},{3\over 2\pi},1)$}
%% WRONG
{$({3\over 2},{3\over 2},1)$}
%% WRONG
{$(1,1,{1\over2})$}
%% WRONG
{$({1\over \sqrt{2}},{1\over \sqrt{2}},1)$}
%% WRONG
{$(0,0,1)$}


%7
%% PROBLEM
Let $R$ be the region of points $(x,y)$ which satisfy
${1\over2}x\le y \le 2x$, $1 \le xy \le 2$.
Using the change of variables $u = xy$, $v = {y\over x}$
transform  $\iint_R xy \,dA$ into an iterated integral in the
$uv$-plane. (Hint: You do not have to solve for $x$ and $y$.)
%% CORRECT
{$\int_{1/2}^2\int_1^2{u\over2v}\,du \,dv$}
%% WRONG
{$\int_{1/2}^2\int_1^2 uv \,du\,dv$}
%% WRONG
{$\int_{1/2}^2\int_1^2 2uv^2 \,du \,dv$}
%% WRONG
{$\int_{x/2}^{2x}\int_1^2u\,du \,dv$}
%% WRONG
{$\int_1^2\int_{x/2}^{2x}\left(uv + {u\over v}\right)\,du\,dv$}

%8
%% PROBLEM
Find a function $f(x,y,z)$ such that 
$\grad f = (y - z\sin(x))\i + (x - z)\j + (\cos(x) - y + 2z)\k$
and then evaluate $f(\pi,2,-1) - f(0,0,0)$.
%% CORRECT
{$2\pi +4$}
%% WRONG
{$\pi + 2$}
%% WRONG
{$0$}
%% WRONG
{$2\pi +3 $}
%% WRONG
{$1$}

%9
%% PROBLEM
Let $C$ be the curve parameterized by
$\r(t) = e^t\i + e^{-t}\j + \k$, $-1\le t \le 1$.
Find the integral that gives the value of the line integral
$\int_C {x^2 + y^2\over z}\,ds$.
%% CORRECT
{$\int_{-1}^1\left(e^{2t} + e^{-2t}\right)^{3/2}\,dt$}
%% WRONG
{$\int_{-1}^1\left(e^{2t} + e^{-2t}\right)\,dt$}
%% WRONG
{$\int_{-1}^1\sqrt{1 + e^{2t} + e^{-2t}}\,dt$}
%% WRONG
{$\int_{-1}^1\sqrt{e^{2t} + e^{-2t}}\,dt$}
%% WRONG
{$\int_{-1}^1 2e^t\left(1 + e^{2t} + e^{-2t}\right)^{3/2}\,dt$}

%10
%% PROBLEM
Let $\F = xy\i + yz\j$ and let $C$ be the curve parameterized by
$\r(t) = t\i + t^2\j + t^3\k,\quad 0\le t \le 1$.
Find the value of
$\int_C\F\bdot d\r$.
%% CORRECT
{$15\over\ds28$}
%% WRONG
{$3\over\ds4$}
%% WRONG
{$7\over\ds24$}
%% WRONG
{$5\over\ds6$}
%% WRONG
{$5\over\ds12$}

%11
%% PROBLEM
Let $C$ be a smooth curve from $(1,1,0)$ to $(0,0,1)$.
Compute $\int_C \cos(yz)\,dx - xz\sin(yz)\,dy - xy\sin(yz)\,dz$.
%% CORRECT
{$-1$}
%% WRONG
{$0$}
%% WRONG
{$\cos(1)$}
%% WRONG
{$2$}
%% WRONG
{$-3\cos(1)$}

%12
%% PROBLEM
Let $C$ be the counterclockwise path around the perimeter of the
triangle with vertices $(0,0)$, $(2,0)$, and $(2,1)$. By Green's
Theorem, the line integral $\int_C e^{x^2}\,dx + e^{y^2}\,dy$.
equals:
%% CORRECT
{$0$}
%% WRONG
{${e\over2}$}
%% WRONG
{$\int_0^2\int_0^{2x}e^{x^2} + e^{y^2}\dydx$}
%% WRONG
{$\int_0^2\int_0^{2x}2xe^{x^2} + 2ye^{y^2}\dydx$}
%% WRONG
{$\int_0^2\int_0^{2x}2xe^{x^2}\i + 2ye^{y^2}\j\dydx$}

%13
%% PROBLEM
Let $\Sigma$ be the upper hemisphere $z = \sqrt{1 - x^2 - y^2}$. The
surface integral $\iint_\Sigma z^2\,dS$ equals:
%% CORRECT
{${2\pi\over3}$}
%% WRONG
{${2\pi}$}
%% WRONG
{${2\over3}$}
%% WRONG
{${1\over3}$}
%% WRONG
{${\pi\over3}$}

%14
%% PROBLEM
Let $\F = x^2y\i - z^2\j + y\k$. Compute $\div\F$.
%% CORRECT
{$2xy$}
%% WRONG
{$2xy - 2z + 1$}
%% WRONG
{$2xy\i - 2z\j + \k$}
%% WRONG
{$2xy + x^2 - 2z + 1$}
%% WRONG
{$(2xy + x^2)\i - 2z\j$}

%15
%% PROBLEM
Let $\F = x^3\i + y^3\j + z^3\k$. Compute $\curl\F$.
%% CORRECT
{$\0$}
%% WRONG
{$3x^2 + 3y^2 + 3z^2$}
%% WRONG
{$3x^2\i + 3y^2\j + 3z^2\k$}
%% WRONG
{$(3y^2 - 3x^2)\k$}
%% WRONG
{$6x + 6y + 6z$}

%16
%% 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
{$\grad\div\F$}

%% END
\bye
