\magnification=\magstep1
\vsize=7.5in
%\voffset=-.25in

\input epsf
\def\r{{\bf r}}
\def\F{{\bf F}}
\def\i{{\bf i}}
\def\j{{\bf j}}
\def\k{{\bf k}}
\def\R{{\bf R}}
\def\0{{\bf 0}}
\def\C{{\cal C}}

\def\grad{{\rm grad\,}}
\def\curl{{\rm curl\,}}
\def\div{{\rm div\,}}
\def\iint{\int\!\!\!\int}
\def\iiint{\int\!\!\!\int\!\!\!\int}

\def\problemskip{\vskip0.5in}

%% TITLE
\noindent
{\bf MATH 225: Calculus III}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam III}\quad{\sl November 29, 2001}\hfill
Instructor:\hbox to 2.5truein{\hrulefill}

\vskip0.5truein
\noindent
Record your answers to the multiple choice problems by placing an $\times$
through one letter for each problem on this page. There are 8 multiple
choice questions worth 5 points each and 4 partial credits problems worth
10 points each. You start with 20 points.  On the partial credit problems
{\it you must show your work and all important steps to receive credit}.

\medskip
You may use a calculator if you wish.

\vskip0.5truein

\def\answerskip{\vskip0.25in}

%1
%% PROBLEM
Let $E$ be the tetrahedron with vertices $(0,0,0)$, $(1,0,0)$, $(0,2,0)$,
and $(0,0,2)$. Express the integral $\iiint_E f(x,y,z)\,dV$ as an iterated
integral.

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

%2
%% PROBLEM
Let $E$ be the region  between the spheres $x^2 + y^2 + z^2 = z$ and
$x^2 + y^2 + z^2 = 2z$.  Which of the following represents $\iiint_E
(x^2 + y^2) \ dV$ in spherical coordinates.

%% CORRECT
$\int_{0}^{2\pi}\int_{0}^{\pi/2}
\int_{\cos(\phi)}^{2\cos(\phi)} \rho^4 \sin^3(\phi)\,d\rho\,d\phi\,d\theta$
%% WRONG
$\int_{0}^{2\pi}\int_{0}^{\pi/2}
\int_{\cos(\phi)}^{2\cos(\phi)}\rho^2\sin(\phi)\,d\rho\,d\phi\,d\theta$
%% WRONG
$\int_{0}^{2\pi}\int_{0}^{\pi/2}\int_1^2
\rho^4\sin(\phi)\,d\rho\,d\phi\,d\theta$
%% WRONG
$\int_{0}^{2\pi}\int_{0}^{\pi/2}\int_{\cos(\theta)}^{2\cos(\theta)} 
\rho^4\sin^3(\theta)\,d\rho\,d\phi\,d\theta$
%% WRONG
$\int_{0}^{2\pi}\int_{-\pi/2}^{\pi/2}\int_{\cos(\phi)}^{2\cos(\phi)} 
\rho^4\sin^3(\phi)\,d\rho\,d\phi\,d\theta$

%3
%% PROBLEM
Let $\C$ be the curve parameterized by $\r(t) = \langle t, t^2,
t^2\rangle$, $0\le t \le 1$.  Evaluate the line integral $\int_{\C} x\,ds$.

%% CORRECT
13/12
%% WRONG
91/3
%% WRONG
0
%% WRONG
1/2
%% WRONG
$(5^{3/2}-1)/24$

%4
%% PROBLEM
Let  $\F(x,y) = (x + y)\i + (y-x)\j$. Calculate $\int_{\C} \F\cdot d\r$
where $\C$ is the circle $\r(t) = \langle\cos(t),\sin(t)\rangle$, $0 \le t
\le 2\pi$.

%% CORRECT
$-2\pi$
%% WRONG
0
%% WRONG
$3\pi/2$
%% WRONG
$-1$
%% WRONG
$\pi$

%5
%% PROBLEM
Let $\F$ be a vector field defined on $\R^3$. Determine which of the
following conditions guarantees that $\F = \nabla f$ for some function $f$.

%% CORRECT
$\curl\F = \0$
%% WRONG
$\div\F = 0$
%% WRONG
$\div\curl\F = 0$
%% WRONG
$\curl\curl\F = \0$
%% WRONG
$\grad\div\F = \0$

%6
%% PROBLEM
Let $\F= (e^x+yz)\i + (e^y-xz)\j + (\sin(z)+x-y)\k$. Compute $\div\F$.

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

%7
%% PROBLEM
Determine which of the following integrals gives the area  of the surface
parameterized by $\r(u,v) = u^2\i + uv\j + v^2\k$, $0 \le u \le 1$, $0\le v
\le 2$.

%% CORRECT
$\int_0^2\int_0^1 2\sqrt{u^4 + 4 u^2 v^2 + v^4}\,du\,dv$
%% WRONG
$\int_0^2\int_0^1 \sqrt{u^4 + u^2 v^2 + v^4}\,du\,dv$
%% WRONG
$\int_0^2\int_0^1 u^4 + u^2 v^2 + v^4\,du\,dv$
%% WRONG
$\int_0^2\int_0^1 \sqrt{2}(u^2 + 4u v + v^2)\,du\,dv$
%% WRONG
$\int_0^2\int_0^1 2(u^2 + v^2)\,du\,dv$

%8
%% PROBLEM
Determine which of the following plots is the surface parameterized by
$$\r(u,v) = u \cos(v)\i + u\sin(v)\j + v\k,\quad -1\le u \le 1, 0 \le
v \le 2\pi$$

%% CORRECT
\epsfysize=2in \epsfbox{exam3.1.eps}
%% WRONG
\epsfysize=1.5in \epsfbox{exam3.2.eps}
%% WRONG
\epsfysize=1.5in \epsfbox{exam3.3.eps}
%% WRONG
\epsfysize=1.5in \epsfbox{exam3.4.eps}
%% WRONG
\epsfysize=1.5in \epsfbox{exam3.5.eps}

%% END

\vfill\eject

\noindent\llap{9.\ }
Let $E$ be the region between the paraboloids $z=x^2 + y^2$ and $z= 2 -
x^2 - y^2$.  Compute  $\iiint_E \sqrt{x^2 + y^2}\,dV$.

\vfill\eject

\noindent\llap{10.\ }
Let $\F = (e^y + yz)\i + x(e^y + z)\j + (xy - 2)\k$.

\medskip

a) Find a function $f$ such that $\F = \nabla f$.

\vfill

b) Use the Fundamental Theorem for Line Integrals to evaluate
$$\int_\C \F\cdot d\r$$
where $\C$ is a curve from $(0,0,0)$ to $(3,1,-1)$.

\vfill\eject

\noindent\llap{11.\ }
Let $R$ be the region bounded by the hyperbolas $xy = 1$ and $xy = 9$ and
the lines $y = x$ and $y=4x$ in the first quadrant. Use the change
of variables $x = uv$, $y = u/v$, $u\ge0$, $v\ge0$, to evaluate
the integral $\iint_R x^2y\,dA$.

\vfill\eject

\noindent\llap{12.\  }
Use Green's Theorem to calculate
$$\int_{C} (e^{x^2} + xy) dx + (x + \sin(y^2)) dy$$
where $C$ is the square with vertices $(0,0)$, $(1,0)$,
$(1, 1)$, $(0, 1)$, oriented counter-clockwise.

\bye
