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\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 26\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 26} 
\bigskip


\itemitem{1.\hskip12pt a)}  Given the following polar coordinates for
a point in $\R^2$, determine Euclidean coordinates that represent the
same point.  Is there more than one such representation?
$$
\left(3,{\pi\over6}\right)\qquad(0,\pi^2)\qquad
\left(-{1\over4},{\pi\over4}\right)\qquad(-8,-37\pi)
$$

\medskip
\itemitem{b)}  Given the following Euclidean coordinates for a point
in $\R^2$, determine polar coordinates that represent the same point.
Is there more than one such representation?
$$
(1,0)\qquad(0,1)\qquad(-2\sqrt3,2)\qquad(-4,-3)
$$

\medskip 
\item{2.}  For a given point in $\R^2$ with polar coordinates
$(r,\theta)$ list all pairs $(r',\theta')$ which determine the same point.

\medskip
\item{3.}  Find the intersections of each of the following pairs of
graphs in polar coordinates.  Use both graphical and algebraic methods
for each pair.
$$
{\eqalign{r&=\sin^2\theta\cr r&=-1\cr}}\qquad\qquad\qquad
{\eqalign{r&=\theta\cr r&=\pi-\theta\cr}}\qquad\qquad\qquad
{\eqalign{\theta&={3\pi\over2}\cr r&=\theta\sin\theta\cr}}
$$

\medskip
\itemitem{4.\hskip12pt a)}  Explain why the area enclosed between the
polar graph of a function $r=f(\theta)$ and the two lines
$\theta=\theta_a$ and $\theta=\theta_b$ is NOT equal to
$\int_{\theta_a}^{\theta_b}f(\theta)\,d\theta$.

\medskip
\itemitem{b)}  If you partition the interval $[\theta_a,\theta_b]$ how
could you estimate the area enclosed between the angles of one
subinterval $[\theta_i,\theta_{i+1}]$ and the graph of $r=f(\theta)$?
Write down a formula.
\itemitem{}  (Hint:  Use a wedge shaped piece of a circle (of what
radius?) to approximate the region over the subinterval
$[\theta_i,\theta_{i+1}]$.  What percent of the area of the whole
circle does this represent?)

\medskip
\itemitem{c)}  Write down the exact area enclosed as a limit of
Riemann sums.  

\medskip
\item{5.}  Sketch the graphs of $r=\cos2\theta$ and
$r={1\over2}+{1\over2}\cos4\theta$.  Find the area bounded by each of
these graphs.

\medskip
\item{7.}  Let $U$ be a bounded region of the plane.  Suppose that you
can determine the length of any cross section of this region through
the origin.  Can you determine the area of $U$?  If so how?  If not,
give an example of two regions with the same lengths of cross sections
through the origin, but which have different areas.






\bye