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