% 15.4: Double Integrals in Polar Form.
\item 
Section 15.4: 

Study why $dA = r dr d\theta$ in polar coordinates.
Practice how to determine the limits of integration
in $\displaystyle{ \int_{\theta_1}^{\theta_2} \int_{f_1(\theta)}^{f_2(\theta)} (\cdots) r dr d\theta }$ 
as in Examples 2-6 of section 15.4.
More specifically:
\begin{enumerate}
\item 
Given a domain $R$ in the $x-y$ plane,
practice drawing $\theta = C$ lines in $R$.
The end points of these lines are lower limit
(the near end point) and upper limit (the far end point)
of the integration $rdr$.
The end points for $d\theta$ are smallest and largest
$C$ among these $\theta = C$ lines.

\item
The end points of the lines
$\theta = C$ must be expressed as
$r = f_1(\theta)$ and $r = f_2(\theta)$.

Given a simple curve $F(x,y)=0$ (such as a line or a circle),
use the substitution $x=r\cos\theta$, $y=r\cos\theta$ to
express it as $r = f(\theta)$.
Examples: $x=1$, $y= -3$, $x+y = 1$, $x^2+y^2 = 4$, etc.
\end{enumerate}

