Evaluating an integral using the Fubini Theorem.

Question

I want to calculate $\int_{B}x^2z$ and for this I have done the following $\int_{B}x^2z=\int_{A}(\rho\sin\phi\cos\theta)^2(\rho\cos\phi)\rho^2\sin\phi=\int_{0}^{\pi/2}\int_{0}^{\pi}\int_{0}^{a}(\rho\sin\phi\cos\theta)^2(\rho\cos\phi)\rho^2\sin\phi d\rho d\phi d\theta$.

Is this argument correct? Thank you very much.

Answer 1

\begin{align}&\int_{0}^{\pi/2}\int_{0}^{\pi}\int_{0}^{a}(\rho\sin\phi\cos\theta)^2(\rho\cos\phi)\rho^2\sin\phi \,d\rho \,d\phi \,d\theta \\ =&\int_{0}^{\pi/2}\int_{0}^{\pi}\int_{0}^{a} \rho^4\sin^3\phi\cos^2\theta\cos\phi \,d\rho \,d\phi \,d\theta\\ =&\int_{0}^{\pi/2}\cos^2\theta\,d\theta \int_{0}^{\pi}\sin^3\phi\cos\phi\,d\phi \int_{0}^{a}\rho^4\,d\rho \end{align} which should be easy to evaluate.