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\centerline{Mathematics 241, section 1 -- Multivariable Calculus}
\centerline{Exam III -- November 26, 2013}
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{\it Directions} 
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Do all work in the blue exam booklet.  There are 100 regular points
and 10 Extra Credit points.
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I.  (20)  Given that 
$$f(x,y) = {y \over x^2 + y^2 + 4},$$
and
$$\eqalign{
f_x = {-2xy\over (x^2 + y^2 + 4)^2}
& \qquad\qquad\qquad f_y = {x^2 - y^2 + 4\over (x^2 + y^2 + 4)^2} \cr
f_{xx} = {2y(3x^2-y^2-4)\over (x^2 + y^2 + 4)^3}\quad 
f_{xy} &= {-2x(x^2-3y^2+4)\over (x^2 + y^2 + 4)^3} \quad
f_{yy} = {-2y(3x^2-y^2+12)\over (x^2 + y^2 + 4)^3}\cr
}$$
find all critical points of $f$ and apply the Second
Derivative Test to each to determine the types of the critical points.
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\noindent
II.  
\item{A)}  (20) Find the maximum and minimum values of 
$$f(x,y) = xy + 2y$$
on the closed and bounded set 
$$S = \{(x,y) \in {\bf R}^2 \mid x^2 + y^2 \le 9\}.$$
\item{B)}  (5)  Give a sketch of the contours of $f$ for the 
maximum and minimum values you found in part A, together with $S$.
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\noindent
III.  All parts of this problem refer to the region
$$R = \{(x,y) \in {\bf R}^2 \mid x^2 + y^2 \le 4, y \ge 0, y \le 2x\}.$$
and the solid $S \subset {\bf R}^3$ with upper boundary $z = x^2$ and 
lower boundary the $xy$-plane.  
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\item{A)} (10) Sketch the region $R$ in the plane.
\item{B)} (10)  Set up iterated integral(s) of the form $\int\int f(x,y)\ dy\ dx$
to compute the volume of $S$.  Do not compute the value.
\item{C)}  (10)  Now set up iterated integral(s) of the form $\int\int f(x,y)\ dx\ dy$
to compute the volume of $S$.  Do not compute the value.
\item{D)}  (10)  Finally, set up a polar coordinate integral that 
will compute the volume of $S$.  Do not compute the value.
\item{E)}  (15)  Finally, compute the volume of $S$ using any one
of your answers from parts B,C,D (your choice).   {\it For full credit, you 
must show all work leading to your final answer.}
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{\it Extra Credit} (10)  
Evaluate the improper double integral
$$\int\int_{{\bf R}^2} e^{1-x^2-y^2}\ dA = \int_{-\infty}^\infty \int_{-\infty}^\infty e^{1-x^2 -y^2}\ dy\ dx.$$
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