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\title{ESP Workshop, Worksheet \#21}
\date{\today}
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\begin{center}
{\bf \Large ESP Workshop, Worksheet \#21}

{\bf \large Tuesday \today}

{\bf \large AI: Eric Katerman}
\end{center}
\vspace{.15in}
\begin{enumerate}
\item Finally... some applications of all this abstract nonsense we've
  been learning over the past few months!  Let's make some real-world
  computations!

\begin{enumerate}
\item Electric charge is distributed over the rectangle $1 \leq x \leq
  3,\ 0 \leq y \leq 2$ so that the charge density at $(x,y)$ is
  \begin{displaymath}
    \sigma (x,y) = 2xy +y^2
  \end{displaymath}
  (measured in coulombs per square meter). Find the total charge on
  the rectangle.
  
\item Find the mass and center of mass of the thin plate (or
  ``lamina'') that occupies the region bounded by $y = e^x, y=0,
  x=0,$ and $x=1$ and has density function $\rho (x,y) = y$.

\item Find the area of the part of the plane $3x +2y +z = 6$
  that lies in the first octant.

\item Find the area of the part of the surface $z = xy$ that lies
  within the cylinder $x^2 +y^2 =1$.

\end{enumerate}

\item It's that time again... time to do some of Professor Durbin's
  old exam problems. 

\begin{enumerate}
\item Suppose that $f$ has continuous first and second partial
  derivatives for $x^2 +y^2 < 9$ and that $f_x (1,2) =0,\ f_y (1,2)
  =0$, and $f_{xx}(1,2) < 0$. Let $d = f_{xx} (1,2) f_{yy} (1,2) -
  [f_{xy} (1,2) ]^2$. What can you conclude about the local extrema of
  $f$ in each of the following cases?
\begin{enumerate}
\item $d<0$
\item $d=0$
\item $d>0$
\end{enumerate}

\item Write (but do not evaluate) a double integral in polar
  coordinates that will give the volume of the solid bounded above by
  the spherical surface $x^2 +y^2 +z^2 =9$ and below by the plane
  $z=2$.

\item Find the minimum distance from the point (0,1) to the parabola
  $y=x^2$.

\item Locate and classify all extreme points of $f(x,y) = 6x^2 -2x^3
  +3y^2 +6xy$.

\item Change the order of integration:
\begin{displaymath}
\int _0 ^2 \int _{x^2} ^{2x} f(x,y) dy dx
\end{displaymath}
Now let $f(x,y) = 3x +5xy$ and evaluate the double integral both ways.
Did you get the same answer?

\item Assume $f(x,y) = 3x +y$ on the region $R = \{ (x,y) : 0\leq x
  \leq 2, 0\leq y \leq 2 \}$. Also assume that $P_1 = \{0, 1/2, 2 \}$,
  $P_2 = \{ 0,1,2 \}$, and $P = P_1 \times P_2$. Find the lower
  approximating sum $L(P)$. How does this compare with the actual
  value of the double integral on $R$?

\item Evaluate the following double integral with $\Omega$ the
  triangle formed by the $x$-axis and the lines $y=x$ and $x=3$.
\begin{displaymath}
\iint _\Omega e^{x+y} dy dx
\end{displaymath}
Check your answer by comparing it with $\iint _\Omega e^{x+y} dx dy$.

\end{enumerate}

\item The figure at the bottom of the page shows the surface created
  when the cylinder $y^2 +z^2 =1$ intersects the cylinder $x^2 +z^2
  =1$. Find the area of this surface.

\item (Math Challenge!) Evaluate the integral
\begin{displaymath}
\int _0 ^1 \int _0 ^1 e^{\max \{ x^2, y^2 \}} dy dx
\end{displaymath}
where $\max \{ x^2, y^2 \}$ means the larger of the numbers $x^2$ and
$y^2$.

\item (Physics Challenge!) A lamina has constant density $\rho$ and
  takes the shape of a disk with center the origin and radius $R$.
  Recall that Newton's Law of Gravitation
\begin{displaymath}
\mathbf{F} = -\frac{GMm}{r^3} \mathbf{r}
\end{displaymath}
(where $\mathbf{F}$ is the gravitational force on the body of mass
$m$, $M$ is the mass of the lamina, $G$ is the gravitational constant,
and $\mathbf{r}$ is the position of the smaller body).

\begin{enumerate}
\item Show that the magnitude of the force of attraction that the
  lamina exerts on a body with mass $m$ located at the point $(0,0,d)$
  on the positive $z$-axis is
\begin{displaymath}
F = 2\pi Gm \rho d \left ( \frac1d - \frac{1}{\sqrt{R^2 +d^2}} \right )
\end{displaymath}
[Hint: Use polar coordinates, and divide the disk into ``polar subrectangles'' and first compute the vertical component of the force exerted by the polar subrectangle $R_{ij}$.]

\item Show that the magnitude of the force of attraction of a lamina
  with density $\rho$ that occupies an entire plane on an object with
  mass $m$ located at a distance $d$ from the plane is $F = 2\pi G m
  \rho$. (Notice that this expression does not depend on $d$!)

\end{enumerate}

\end{enumerate}

\end{document}