\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 December 5, 1991}\hfill Score:\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}

%1
%% PROBLEM
Compute $\int_0^1\int_0^x e^{x^2} \,dy\,dx$.
%% CORRECT
$(e-1)/2$
%% WRONG
$e$
%% WRONG
$0$
%% WRONG
${1\over2} - e$
%% WRONG
$1$


%2
%% PROBLEM
Reverse the order of integration in the integral $\int_1^2\int_1^{x^3}
f(x,y)\,dy\,dx$
%% CORRECT
$\int_1^8\int_{y^{1/3}}^2f(x,y)\,dx\,dy$
%% WRONG
$\int_1^{x^3}\int_1^2f(x,y)\,dx\,dy$
%% WRONG
$\int_1^2\int_{x^3}^1f(x,y)\,dx\,dy$
%% WRONG
$\int_0^8\int_{y^3}^2f(x,y)\,dx\,dy$
%% WRONG
$\int_0^8\int_0^{y^{1/3}}f(x,y)\,dx\,dy$


%3
%% PROBLEM
The area of the region bounded by $y = 9 - x^2$ and the $x$-axis is 36. Find the centroid of this region.

\centerline{\epsfxsize=1.5truein\epsfbox{exam3.3.1.eps}}

%% CORRECT
$(0,3.6)$
%% WRONG
$(0,4.0)$
%% WRONG
$(0,2.4)$
%% WRONG
$(0,3.2)$
%% WRONG
$(0,2.8)$

%4
%% PROBLEM
Compute the area of the region that lies inside the cardioid $r = 1 -
\cos(\theta)$ in the first quadrant.

\centerline{\epsfxsize=1truein\epsfbox{exam3.4.1.eps}}

%% CORRECT
${3\pi\over8} - 1$
%% WRONG
${\pi\over2}$
%% WRONG
${5\pi\over8} - {1\over2}$
%% WRONG
${7\pi\over16} - {1\over4}$
%% WRONG
${9\pi\over16}$



%5
%% PROBLEM
Let $D$ be the solid in the first octant below the plane $4x + 4y + z = 4$. Suppose the density of $D$ is given by $\delta(x,y,z) = 4 - z$. Which of the following integrals gives the total mass of~$D$.

%\centerline{\epsfxsize=1.5truein\epsfbox{exam3.5.1.eps}}

%% CORRECT
$\int_0^1\int_0^{1-x}\int_0^{4 - 4x - 4y} 4 - z\,dz\,dy\,dx$
%% WRONG
$\int_0^4\int_0^{4-z}\int_0^{4 - 4x - 4y} 1\,dz\,dy\,dx$
%% WRONG
$\int_0^1\int_0^{1-x}\int_0^{4 - z} 4 - 4x - 4y\,dz\,dy\,dx$
%% WRONG
$\int_0^1\int_0^{4-x}\int_0^{4 -  4y} 4 - z\,dz\,dy\,dx$
%% WRONG
$\int_0^4\int_0^{4-x}\int_0^{4 - 4x - 4y} 4 - z\,dz\,dy\,dx$


%6
%% PROBLEM
Find the volume of the solid bounded above by $z = 4 - x^2 - y^2$, below by
the $xy$-plane, and on the sides by $x^2 + y^2 = 1$.
%% CORRECT
$7\pi/2$
%% WRONG
$22\pi/3$
%% WRONG
$4\pi$
%% WRONG
$9\pi/2$
%% WRONG
$25\pi/3$


%7
%% 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{exam3.7.1.eps}}

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


%8
%% 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
$\int_1^2\int_{-2}^1{1\over 3} uv\,du\,dv$



%9
%% PROBLEM
Determine which of the following represents the vector field $\F(x,y) =
-{y\over5}\i + {x\over5}\j$.
%% CORRECT
\epsfxsize=1.5truein\epsfbox{exam3.10.1.eps}
%% WRONG
\epsfxsize=1.5truein\epsfbox{exam3.10.2.eps}
%% WRONG
\epsfxsize=1.5truein\epsfbox{exam3.10.3.eps}
%% WRONG
\epsfxsize=1.5truein\epsfbox{exam3.10.4.eps}
%% WRONG
\epsfxsize=1.5truein\epsfbox{exam3.10.5.eps}


%10
%% PROBLEM
Let $\F(x,y) = x^2(x + y)\i + y^2(y + z)\j + z^2(x + z)\k$. Compute $\div
\F$.
%% CORRECT
$3(x^2 + y^2 + z^2) + 2(xy + yz + xz)$
%% WRONG
$(3x^2 + 2xy)\i + (3y^2 + 2yz)\j + (3z^2 + 2xz)\k$
%% WRONG
$xyz(3x + 2y)(3y + 2z)(3z + 2x)$
%% WRONG
$-(y^2\i + z^2\j + x^2\k)$
%% WRONG
$0$


%11
%% PROBLEM
Let $\F(x,y) = e^x\cos(y)\i + e^{-x}\sin(y)\j + \k$. Compute $\curl \F$
%% CORRECT
$(e^x - e^{-x})\sin(y)\k$
%% WRONG
$(e^x + e^{-x})\cos(y)\k$
%% WRONG
$(e^x + e^{-x})\sin(y)$
%% WRONG
$e^x\cos(y)\i + e^{-x}\cos(y)\j$
%% WRONG
$\0$


%12
%% PROBLEM
Calculate the line integral $\int_C y \,ds$ where $C$ is the curve
parameterized by $\r(t) = t\i + 2t\j + t^2\k$, $0 \le t \le 1$.
%% CORRECT
$2.637$
%% WRONG
$1.905$
%% WRONG
$3.872$
%% WRONG
$2.733$
%% WRONG
$3.223$


%13
%% PROBLEM
Let $C$ be parameterized by $\r(t) = (t^3 - 1)\i + t^2\j + t\k$, $0\le t
\le 1$.

Evaluate $\int_C 1\,dx + x\,dy + y\,dz$.

%% CORRECT
$0.733$
%% WRONG
$0.5$
%% WRONG
$0.333$
%% WRONG
$1.2$
%% WRONG
$0.822$


%14
%% PROBLEM
Let $C$ be parameterized by $\r(t) = e^{\cos(t)}\i + e^{\sin(t)}\j$, $0 \le t \le \pi$, and let
$f(x,y) = \sqrt{x^2 + y^2}$. Calculate $\int_C \grad(f)\cdot d\r$.
%% CORRECT
$\sqrt{e^{-2} + 1} - \sqrt{e^2 + 1}$
%% WRONG
$\sqrt{e^{-2} + e^2} - \sqrt{2}$
%% WRONG
$0$
%% WRONG
$e^{-1} - e$
%% WRONG
$2\sqrt{e^{-2} + 1}$



%15
%% PROBLEM
Let $C$ be the boundary of the upper half of a circle of radius~1. Use
Green's Theorem to evaluate $\int_C (x^2 + y^2)\,dx + x y \, dy$.

\centerline{\epsfxsize=2truein\epsfbox{exam3.16.1.eps}}

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


%% END
\bye
