\magnification=\magstep1
\input epsf
\input calc
\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 18, 1996}\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 5 points each. You start
with 25~points.

\def\answerspace{0.25in}

%1
%% PROBLEM
Compute the area of the region that lies inside the cardioid $r = 2 -
\cos(\theta)$ above the $x$-axis.

\centerline{\epsfxsize=2truein\epsfbox{exam3.1.eps}}

%% CORRECT
${9\pi\over4}$
%% WRONG
$2\pi$
%% WRONG
${15\pi\over4}$
%% WRONG
${7\pi\over2}$
%% WRONG
${3\pi\over2}$


%2
%% PROBLEM
Let $D$ be the solid in the first octant below the plane $2x + 2y + z =
2$. Suppose the density of $D$ is given by $\delta(x,y,z) = 2 - z$.
Which of the following integrals gives the total mass of~$D$.

%% CORRECT
$\int_0^1\int_0^{1-x}\int_0^{2 - 2x - 2y} 2 - z\,dz\,dy\,dx$
%% WRONG
$\int_0^2\int_0^{2-z}\int_0^{2 - 2x - 2y} 1\,dz\,dy\,dx$
%% WRONG
$\int_0^1\int_0^{1-x}\int_0^{2 - z} 2 - 2x - 2y\,dz\,dy\,dx$
%% WRONG
$\int_0^1\int_0^{2-x}\int_0^{2 -  2y} 2 - z\,dz\,dy\,dx$
%% WRONG
$\int_0^2\int_0^{2-x}\int_0^{2 - 2x - 2y} 2 - z\,dz\,dy\,dx$


%3
%% PROBLEM
Find the volume of the solid bounded above by the paraboloid $ x^2 +
y^2 + z = 4$ and below by the $xy$-plane.

\centerline{\epsfxsize=1.5truein\epsfbox{exam3.3.eps}}

%% CORRECT
$8\pi$
%% WRONG
${32\pi\over3}$
%% WRONG
$16\pi$
%% WRONG
${9\pi\over2}$
%% WRONG
${25\pi\over3}$


%4
%% PROBLEM
Let $D$ be the solid bounded above by $\rho = 2\sin(\phi)$ and below by
$\phi = \pi/2$ in spherical coordinates. Compute $\iiint_D z\,dV$.

\centerline{\epsfxsize=4truein\epsfbox{exam3.4.eps}}

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


%5
%% PROBLEM
Let $R$ be the region bounded by the lines
$$\matrix{x+y = 1,&x+y = 2\cr
x - 2y = -2,&x - 2y = 1\cr}$$
Determine which of the following integrals gives the value of
$\iint_Rx y \,dA$ after the substitution
$u = x+y$, $v = x - 2y$.

%% CORRECT
$\int_{-2}^1\int_1^2{1\over 27} (u - v)(2u + v)\,du\,dv$
%% WRONG
$\int_{-2}^1\int_1^2{1\over 9} (v - u)(2u + v)\,du\,dv$
%% WRONG
$\int_{-2}^1\int_1^2(x+y)(x-2y)\,dx\,dy$
%% WRONG
$\int_1^2\int_{-2}^1{1\over3}(x+y)(2y-x)\,dy\,dx$
%% WRONG
$\int_1^2\int_{-2}^1{1\over 3} uv\,du\,dv$


%6
%% PROBLEM
Determine which of the following represents the vector field $\F(x,y) =
-{y\over5}\i + {x\over5}\j$.

%% CORRECT
\epsfxsize=1.5truein\epsfbox{exam3.6.1.eps}
%% WRONG
\epsfxsize=1.5truein\epsfbox{exam3.6.2.eps}
%% WRONG
\epsfxsize=1.5truein\epsfbox{exam3.6.3.eps}
%% WRONG
\epsfxsize=1.5truein\epsfbox{exam3.6.4.eps}
%% WRONG
\epsfxsize=1.5truein\epsfbox{exam3.6.5.eps}


%7
%% PROBLEM
Let $\F(x,y) = (x^3 + yz)\i + (y^3 + xz)\j + (z^3 + xy)\k$. Compute $\div \F$.

%% CORRECT
$3(x^2 + y^2 + z^2)$
%% WRONG
$(3x^2 + y+z)\i + (3y^2 + x+z)\j + (3z^2 + x+y)\k$
%% WRONG
$3x^2\i + 3y^2\j + 3z^2\k$
%% WRONG
$3(x^2 + y^2 + z^2) + 2(x + y + z)$
%% WRONG
$0$


%8
%% PROBLEM
Let $\F(x,y) = e^{-y}\cos(x)\i + e^{-y}\sin(x)\j + z\k$. Compute $\curl \F$.

%% CORRECT
$ 2e^{-y}\cos(x)\k$
%% WRONG
$-2e^{-y}\sin(x)\k$
%% WRONG
$1-2e^{-y}\sin(x)$
%% WRONG
$-e^{-y}\cos(x)\i - e^{-y}\cos(x)\j$
%% WRONG
$\0$


%9
%% PROBLEM
Calculate the line integral $\int_C x \,ds$ where $C$ is the curve
parameterized by $\r(t) = t\i + t^2\j + t\k$, $0 \le t \le 1$.

%% CORRECT
$0.989$
%% WRONG
$2.637$
%% WRONG
$1.905$
%% WRONG
$3.872$
%% WRONG
$3.223$


%10
%% PROBLEM
Let $C$ be the space curve parameterized by $\r(t) = (t^2 - 1)\i + t^3\j
+ (t-1)\k$, $0\le t \le 1$. Evaluate $\int_C z\,dx + x\,dy + y\,dz$.

%% CORRECT
$-0.483$
%% WRONG
$-0.733$
%% WRONG
$-0.5$
%% WRONG
$-1.235$
%% WRONG
$-0.822$


%11
%% PROBLEM
Let $C$ be the plane curve parameterized by $\r(t) = \sqrt{1+\cos(t)}\i
+ \sqrt{1+\sin(t)}\j$, $0 \le t \le \pi$, and let $f(x,y) = (x^2 +
y^2)^{x^2y^2}$. Calculate $\int_C \grad(f)\cdot d\r$.

%% CORRECT
$-8$
%% WRONG
$\sqrt{2} - 1$
%% WRONG
$0$
%% WRONG
$-\sqrt{2}$
%% WRONG
$4$


%12
%% PROBLEM
Let $C$ be the boundary of the upper half of a circle of radius~1. Use
Green's Theorem to evaluate $\int_C   x y \, dx + (x^2 + y^2)\,dy$

\centerline{\epsfxsize=2truein\epsfbox{exam3.12.eps}}

%% CORRECT
$0$
%% WRONG
$-1/3$
%% WRONG
$-1/2$
%% WRONG
$-3/4$
%% WRONG
$-2/3$

%13
%% PROBLEM
Evaluate the iterated integral
$\int_0^{\sqrt{\pi}}\int_0^z\int_0^{yz} \cos(y^2)\,dx\,dy\,dz$.

%% CORRECT
${1\over2}$
%% WRONG
${\pi\over2}$
%% WRONG
$\sqrt{\pi}$
%% WRONG
${1\over4}$
%% WRONG
${\pi-1\over2}$


%14
%% PROBLEM
Compute $\iint_R e^{-x^2-y^2}\,dA$, where $R$ is the region defined by
$x^2 + y^2 \le 2$.

%% CORRECT
$\pi(1-e^{-2})$
%% WRONG
$2\pi e^{-2}$
%% WRONG
$4\pi$
%% WRONG
$2\pi(e^{-2}-1)^2$
%% WRONG
$(1-e^{-2})/2$


%15
%% PROBLEM
Find the average value of the function $f(x,y) = x y^2 - z x^2$ in the
solid region defined by $0\le x \le 2$, $0\le y\le 3$, and $0\le z \le 1$.

%% CORRECT
7/3
%% WRONG
1/3
%% WRONG
5/6
%% WRONG
3/2
%% WRONG
11/6

%% END
\bye
