\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 December 1, 1992}\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 15 multiple choice questions worth 6 points each. You start with 10 points.

\def\answerspace{0.25in}


%% PROBLEM
Compute $\int_1^e\int_0^{1/y} \ln(y) \, dx\,dy$.

%% CORRECT
$1\overd2$
%% WRONG
$\ln(2)$
%% WRONG
$0$
%% WRONG
$1$
%% WRONG
$2$



%% PROBLEM
Reverse the order of integration in the integral
$\int_{-\sqrt{2}}^{\sqrt{2}}\int_1^{3-x^2} f(x,y)\,dy\,dx$.

%% CORRECT
$\int_1^3\int_{-\sqrt{3-y}}^{\sqrt{3-y}}f(x,y)\,dx\,dy$
%% WRONG
$\int_1^{3-x^2}\int_{-\sqrt{2}}^{\sqrt{2}}f(x,y)\,dx\,dy$
%% WRONG
$\int_{-\sqrt{2}}^{\sqrt{2}}\int_1^{3-y^2} f(x,y)\,dx\,dy$.
%% WRONG
$\int_0^3\int_0^{3-y^2}f(x,y)\,dx\,dy$
%% WRONG
$\int_{-2}^2\int_{-\sqrt{3-y}}^{\sqrt{3-y}}f(x,y)\,dx\,dy$

%% PROBLEM
Find the $y$-coordinate of the centroid of the plane region bounded
by $y = 2x$ and $y = x^2$.

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

%% CORRECT
$1.6$
%% WRONG
$2.4$
%% WRONG
$2.2$
%% WRONG
$1.8$
%% WRONG
$2.6$


%% PROBLEM
Calculate the area between the spirals $r = \theta$ and $r = 2\theta$
above the $x$-axis.

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

%% CORRECT
$\pi^3/2$
%% WRONG
$\pi^2/2$
%% WRONG
$\pi^2$
%% WRONG
$\pi^3/6$
%% WRONG
$2\pi$


%% PROBLEM
Evaluate $\iint_R e^{-x^2-y^2}\,dA$ where $R$ is the region $x^2 + y^2 \le 1$.

%% CORRECT
$\pi(1-e^{-1})$
%% WRONG
$2\pi\,e^{-1}$
%% WRONG
$(1-e^{-1})/2$
%% WRONG
$\pi(e - 1)^{-1}$
%% WRONG
$\pi$


%% PROBLEM
Evaluate the iterated integral $\int_0^{\sqrt{\pi/2}}\int_0^y\int_0^{xy}
\sin(x^2)\,dz\,dx\,dy$.

%% CORRECT
$(\pi-2)/8$
%% WRONG
$\pi/4$
%% WRONG
$\sqrt{2\pi}$
%% WRONG
$(\pi - 1)/4$
%% WRONG
$\sqrt{\pi}$


%% PROBLEM
Determine which of the following sets of inequalities describes
the solid region bounded above by the sphere $x^2 + y^2 + z^2 = 80$
and below by the paraboloid $2z = x^2 + y^2$.

%% CORRECT
\enspace\vtop{\halign to 1.5truein{\hfil$#$&\  $#$\hfil\cr
{1\overd 2}r^2 \le &z \le \sqrt{80 - r^2}\cr
0 \le &r \le 4\cr
0 \le &\theta \le 2\pi\cr}}

%% WRONG
\enspace\vtop{\halign to 1.5truein{\hfil$#$&\  $#$\hfil\cr
0 \le &\rho \le \sqrt{80 }\cr
0 \le &\phi \le \pi/6\cr
0 \le &\theta \le 2\pi\cr}}

%% WRONG
\enspace\vtop{\halign to 1.5truein{\hfil$#$&\  $#$\hfil\cr
0 \le &z \le \sqrt{80 - r^2}\cr
0 \le &r \le 8\cr
0 \le &\theta \le \pi\cr}}

%% WRONG
\enspace\vtop{\halign to 1.5truein{\hfil$#$&\  $#$\hfil\cr
{1\overd 2}r \le &\rho \le \sqrt{80}\cr
0 \le &\phi \le \pi/3\cr
0 \le &\theta \le \pi\cr}}

%% WRONG
\enspace\vtop{\halign to 1.5truein{\hfil$#$&\  $#$\hfil\cr
{1\overd 2}r^2 \le &z \le \sqrt{80}\cr
0 \le &r \le \sqrt{40}\cr
0 \le &\theta \le 2\pi\cr}}


%% PROBLEM
Let $D$ be the tetrahedron bounded by the planes $2x + y + z = 10$,
$x - 2y = 0$, $x = 3$ and $z = 0$. Determine which of the
following integrals gives $\iiint_D f(x,y,z)\,dV$.

%% CORRECT
$\int_3^4\int_{x/2}^{10-2x}\int_0^{10-y-2x}f(x,y,z)\,dz\,dy\,dx$
%% WRONG
$\int_0^3\int_0^{10-2x}\int_0^{x/2}f(x,y,z)\,dz\,dy\,dx$
%% WRONG
$\int_0^3\int_{2y}^{(10-y)/2}\int_0^{10-y-2x}f(x,y,z)\,dz\,dx\,dy$
%% WRONG
$\int_3^5\int_{x/2}^{10-2x}\int_{2y}^{10-y-2x}f(x,y,z)\,dz\,dy\,dx$
%% WRONG
$\int_0^4\int_0^{10-2x}\int_0^{10-y}f(x,y,z)\,dz\,dy\,dx$


%% PROBLEM
Let $D$ be the region cut from the solid sphere of radius $a$ by
the veritcal planes $y = x$ and $y = 0$ in the first octant. Suppose
the density of $D$ is given by $\delta(x,y,z) = b z$. Compute the
total mass of $D$ ($a$ and $b$ are constants).

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

%% CORRECT
${\pi a^4 b\overd 32}$
%% WRONG
${\pi a^3 b\overd 8}$
%% WRONG
${\pi a^3 b\overd 4}$
%% WRONG
${\pi a^2 b^2\overd 16}$
%% WRONG
${\pi a^3 b^2\overd 2}$


%% PROBLEM
Let $R$ be the region bounded by the lines $y = x$, $y = 4x$, $y
= 1/x$, and $y = 4/x$. Use the transformation $x = uv$, $y = u/v$
to rewrite the integral $\iint_R\sqrt{xy}\,dA$ as an integral over
a region in the $uv$-plane.

$$\epsfxsize=2truein\epsfbox{exam3.4.eps}$$

%% CORRECT
$\int_{1/2}^1\int_1^2 {2u^2\overd v}\,du\,dv$
%% WRONG
$\int_1^2\int_1^2 u\,du\,dv$
%% WRONG
$\int_{1/4}^1\int_1^4 -{2u\overd v}\,du\,dv$
%% WRONG
$\int_0^1\int_1^4 {u^2\overd v}\,du\,dv$
%% WRONG
$\int_1^4\int_1^4 -{u\overd 2v}\,du\,dv$


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

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


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

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


%% PROBLEM
Let $\CC$ be the directed line segment from the point $(1,2)$ to
the point $(5,3)$. Compute the value of line integral $\int_\CC xy\,ds$.

%% CORRECT
$47\sqrt{17}\overd 6$
%% WRONG
$221\overd 2$
%% WRONG
$30$
%% WRONG
$23\sqrt{13}\overd 2$
%% WRONG
$331\overd 3$


%% PROBLEM
Compute the flow integral $\int_\CC\F\cdot d\r$ of the velocity
field $\F(x,y,z) = y\i - x\j + \k$ along the helix
$\r(t) = \cos(\pi t)\i + \sin(\pi t)\j + t\k$, $0 \le t \le 4$.

%% CORRECT
$4(1-\pi)$
%% WRONG
$(\pi-1)/4$
%% WRONG
$2\pi$
%% WRONG
$\pi/4$
%% WRONG
$0$


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

%% CORRECT
$12$
%% WRONG
$10$
%% WRONG
$8$
%% WRONG
$6$
%% WRONG
$4$

%% END
\end
