Q 21. - Calculus

Question

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

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

Step-by-Step Solution

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 1: Given Information

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

Step 2: Simplification

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

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