How to find the spherical coordinates limits calculate the volume of the solid region?

Question

The solid bounded below by the hemisphere $\rho=1,$ $ 0\leq z$ and above by the cardoid of revolution $\rho=1+\cos\phi$ I am new to the triple integral in spherical coordinates. I know that limits of $\rho$ will be $1\leq\rho\leq 1+\cos\phi$, the limits on $\theta$ will be $0\leq\theta\leq2\pi$. So the integral will be $$\int_{0}^{2\pi}\int_{?}^{?}\int_{1}^{1+\cos\phi}(\rho)^2\sin\phi d\rho d\phi d\theta$$ But how to find the limits of $\phi$ in this case, I don't know. Can anyone help! Thanks in advance!

Answer 1

Notice that for the cardioid of revolution, $\rho\ge1$ for $0\le\phi\le\pi/2$ and $\rho<1$ for $\pi/2<\phi\le\pi$. It intersects the hemisphere only in the $xy$ plane. Therefore, $0\le\phi\le\pi/2$.