Question

Set up the appropriate triple integral with spherical coordinates to show that

$M_{XZ}=\frac{1}{16}\mathrm{\pi k}$

Short answer

$M_{XZ}={\int }_0^{\frac{\mathrm{\pi }}{2}}{\int }_0^{\frac{\mathrm{\pi }}{2}}{\int }_0^{1}k{\rho }^3{\sin }^2\varphi \sin \theta d\rho d\theta d\varphi$

Step-by-step solution

Given Information

We need to setup the an expression equivalent to$M_{XZ}=\frac{1}{16}\mathrm{\pi k}$involving spherical coordinates.

Simplification

The appropriate triple integral with spherical coordinates in this case will be

$M_{XZ}={\int }_0^{\frac{\mathrm{\pi }}{2}}{\int }_0^{\frac{\mathrm{\pi }}{2}}{\int }_0^{1}k{\rho }^3{\sin }^2\varphi \sin \theta d\rho d\theta d\varphi$