Find the volume of the solid enclosed by the surface $\rho=1+\cos\phi$

Question

Find the volume of the solid enclosed by the surface $\rho=1+\cos\phi$. How do I set up the triple integral for this? I know that $1+\cos\theta$ is a cardioid but how would it look like as a surface?

Answer 1

The volume is found by integrating $\iiint 1\cdot dV$. The real trick is in finding the bounds. The easiest way to do this would be to set up the integral order with the angle first:

$$V = \int_0^{2\pi}\int_0^2 \int_0^{\cos^{-1}(\rho-1)} \rho^2\sin\phi \:d\phi d\rho d\theta $$