Area calculation

Question

How could we best approach calculating the area inside $r=\cos^{2n-1}(x)+\sin(x)$, $0\leq x\leq \pi$, for $n=1,2,...$?

For $n=3$ we get the following "potato/bean" graph:

and for $n=51$ we get

Answer 1

You should rewrite that with the standard polar coordinate variables:

$$r=\cos^{2n-1}(\theta)+\sin(\theta),\quad 0\leq \theta\leq \pi$$

Then use the standard formula for area of polar coordinates:

$$A=\frac 12\int_a^b (r(\theta))^2\,d\theta$$

Do you need help going farther?