## Integration in Polar Coordinates

We can use all the normal techniques of integration when integrating in polar coordinates. Sup[pose
$\frac{dr}{d \theta } = \frac{a^2}{r^2} cos \theta$
, with
$r( \pi .4 )=a$
.
Separating variables gives
$a^2 d = cos \theta d \theta$
and then integrating
$\int r^2 dr = \int a^2 cos \theta d \theta \rightarrow \frac{r^3}{3} = \frac{a^2 sin 2 \theta}{2} +c$
.
Substitute
$r( \pi .4)=a$
.
$\frac{a^3}{3}=\frac{a^2 sin 2( \pi /4)}{2}+c \rightarrow c= \frac{a^3}{3}- \frac{a^2 sin (\pi /2)}{2}=\frac{a^3}{3}- \frac{a^2}{2}$
.
The equation of the curve is
$r^3=3(\frac{a^2 sin 2 \theta }{2} +\frac{a^3}{3}- \frac{a^2}{2} )$
.