Volume of sphere $x^2+y^2+z^2 =a^2$ and over lemniscate $r^2=a^2\cos2\theta$

Question

Volume of sphere $x^2+y^2+z^2 =a^2$ and over lemniscate $r^2=a^2\cos2\theta$

Now I am having trouble finding limits of integration.

I know limits for $z$ are $ \pm (a^2 - (x^2 + y^2))$ and limits for $r$ are $\pm (a \sqrt{\cos2\theta})$.

How do I find limits of $\theta$ ?

Thanks

Answer 1

Hint:

In order to find the limits for $\theta$ notice that $\theta\in(-\pi/4,\pi/4)$ gives a curve in the right half plane. Also $\theta \in(\pi/4, 3\pi/4)$ gives a curve in the left half plane. Maybe drawing the graph can help you.