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

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!

## 1 Answer

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, $$0\le\phi\le\pi/2$$.

• Sir the limits on the $\rho$ and $\theta$ I have written are correct? – Noor Aslam Jan 8 at 12:47
• @NoorAslam They are correct – Shubham Johri Jan 8 at 18:01