Someone knows the Area and Perimeter of the family of polar curves: r(a)= n + m*sin(a)^2.?

Question

Someone knows the Area and Perimeter of the family of polar curve:

$$r(a)= n + m\cdot \sin^{2}(a)$$

Where $(n,m) \in R$. And $a\in (0,2\pi)$

What I need is a closed-form on $(n,m)$.

Thanks.

Answer 1

If $m$ and $n$ are such that $r(\phi):=n+m\sin^2\phi>0$ for all $\phi$ then you can use the following formula: $${\rm enclosed\ area}={1\over2}\int_0^{2\pi}r^2(\phi)\>d\phi\ .$$ The result will of course be an expression containing the parameters $m$ and $n$.