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  
\[r^3=3(\frac{a^2 sin 2 \theta }{2} +\frac{a^3}{3}- \frac{a^2}{2} )\]
.

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