\input calc
\input epsf
\def\problemspace{0.75in}
\def\C{{\cal C}}

%% TITLE
\noindent
{\bf Math 225: Calculus III}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam III}\quad{\sl Apr. 14, 1994}\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
Find the area of the region enclosed by the curve
$r = \sqrt{3 - 2\sin(2\theta)}$.
$$\epsfxsize=2truein\epsfbox{exam3.1.eps}$$
%% CORRECT
$3\pi$
%% WRONG
$2\pi$
%% WRONG
$5\pi/3$
%% WRONG
$7\pi/3\pi$
%% WRONG
$8\pi/3$


%% PROBLEM
Evaluate $\int_0^1\int_0^{\sqrt{x}}\int_0^y e^{-x^2}\,dz\,dy\,dx$
%% CORRECT
$(1-e^{-1})/4$
%% WRONG
$1/4$
%% WRONG
$e^{-1}/2$
%% WRONG
$(e-1)/2$
%% WRONG
$(e^{-1} - 1)/2$


%% PROBLEM
Find the volume of the solid region in the first octant bounded by the
plane $x + y = 1$ and the surface $z = \sin(\pi x)$.
%% CORRECT
$1/\pi$
%% WRONG
$\pi$
%% WRONG
$1/2$
%% WRONG
$\pi/2$
%% WRONG
$1$


%% PROBLEM
Let $R$ be the region defined by $1 \le x \le 2$, $1\le x y \le 2$. Using
the substitution $u = x$, $v = x y$, transform $\iint_R y\,dA$ into an
iterated integral in the $uv$-plane.
%% CORRECT
$\int_1^2\int_1^2 {v\over u^2}\,du\,dv$
%% WRONG
$\int_1^2\int_1^{2/u} {v\over u}\,du\,dv$
%% WRONG
$\int_1^2\int_1^{2v/u} {1\over u^2}\,du\,dv$
%% WRONG
$\int_1^2\int_1^2 v\,du\,dv$
%% WRONG
$\int_1^2\int_1^2 {v\over u}\,du\,dv$


%% PROBLEM
Let $D$ be the solid region below the paraboloid $z = 4 - x^2 - y^2$ and
above the $xy$-plane. The density of $D$ is given by $\delta(x,y,z) = 4 -
z$. Given that the total mass of $D$ is $64\pi/3$, compute the center of
mass of $D$.
%% CORRECT
$(0,0,1)$
%% WRONG
$(0,0,7/8)$
%% WRONG
$(0,0,3/4)$
%% WRONG
$(0,0,5/4)$
%% WRONG
$(0,0,3/2)$


%% PROBLEM
Find the volume of the solid region between the cone $z^2 = 4(x^2 + y^2)$
and the paraboloid $z = 1 + x^2 + y^2$.
$$\epsfxsize=1.5truein\epsfbox{exam3.2.eps}$$
%% CORRECT
$\pi/6$
%% WRONG
$\pi$
%% WRONG
$3\pi/4$
%% WRONG
$\pi/2$
%% WRONG
$\pi/3$


%% PROBLEM
Find the average value of $f(x,y,z) = (x^2 + y^2 + z^2)^{1/2}$ over the
solid unit ball, $x^2 + y^2 + z^2 \le 1$.
%% CORRECT
$3/4$
%% WRONG
$1/2$
%% WRONG
$5/8$
%% WRONG
$7/8$
%% WRONG
$2/3$


%% PROBLEM
Find the vector field represented in the following plot:
$$\epsfxsize=2truein\epsfbox{exam3.3.eps}$$
%% CORRECT
${1\over2\sqrt{x^2+1}}\i + {x\over2\sqrt{x^2+1}}\j$
%% WRONG
${x\over2\sqrt{x^2+y^4}}\i + {y^2\over2\sqrt{x^2+y^4}}\j$
%% WRONG
${x\over2\sqrt{x^2+1}}\i - {x^2\over2\sqrt{x^4+1}}\j$
%% WRONG
$-{y\over10}\i + {x\over 10}\j$
%% WRONG
${x\over10}\i + {y\over 10}\j$


%% PROBLEM
Let $\F = z\cos((x+y)z)\i + z\cos((x+y)z)\j + (x + y)\cos((x+y)z)\k$.
Compute $\div\F$.

%% CORRECT
$-((x+y)^2+2z^2)\sin((x+y)z)$
%% WRONG
$0$
%% WRONG
$-z^2\sin((x+y)z)\i-z^2\sin((x+y)z)\j - (x+y)^2\sin((x+y)z)\k$
%% WRONG
$(x+y)^2\sin((x+y)z)$
%% WRONG
$yz\cos((x+y)z) - xz\sin((x+y)z)$


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


%% PROBLEM
Let $\C$ be the curve defined by $2y = 1 - x^2$, $0 \le
x \le 1$.  Compute the value of the line integral $\int_\C x\,ds$.

%% CORRECT
$(2\sqrt{2}-1)/3$
%% WRONG
$1/2$
%% WRONG
$(\sqrt{3}-1)/2$
%% WRONG
$1/4$
%% WRONG
$7/3$


%% PROBLEM
Let $\F$ be the velocity field $x^2 y\i - x z^2\j + y^2\k$ and let $\C$ be
the curve parameterized by $\r(t) = t\i + t^2\j - 2t\k$, $0 \le t \le 1$.
Compute the flow integral $\int_\C \F\bdot d\r$.
%% CORRECT
$-9/5$
%% WRONG
$-7/3$
%% WRONG
$-7/2$
%% WRONG
$-9/4$
%% WRONG
$-11/6$


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


%% PROBLEM
Let $\C$ be the boundary of the region inside the unit circle in the first
quadrant (oriented counterclockwise). Use Green's Theorem to evaluate
$\int_\C x^2y\,dx - xy^2\,dy$.
$$\epsfxsize=1truein\epsfbox{exam3.4.eps}$$
%% CORRECT
$-\pi/8$
%% WRONG
$\pi$
%% WRONG
$-\pi/3$
%% WRONG
$\pi/4$
%% WRONG
$-3\pi/4$


%% PROBLEM
Determine which of the following formulas is true for an arbitrary function
$f$ or vector field $\F$.
%% CORRECT
$\div\curl\F = 0$
%% WRONG
$\div\grad f = 0$
%% WRONG
$\curl\curl\F = \0$
%% WRONG
$\curl\div\F = \0$
%% WRONG
$\grad\div\F = \0$

%% END
\end
