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• Notes

Use spherical coordinates to Evaluate $\iiint_{B}\left(x^{2}+y^{2}+z^{2}\right)^{2} d V$
where $B$ is the ball with center the origin and radius $5 .$

$\iiint \left(x^{2}+y^{2}+z^{2}\right)^{2} d V \rightarrow d x d y d z$

$\left(x^{2}+y^{2}+z^{2}\right)^{2}= \rho^{2}$

$\int_{0}^{\pi} \int_{0}^{2 \pi} \int_{0}^{5}\left(\rho^{4}\right) * \rho^{2} \sin \phi d \rho d \phi d \theta$

$\int_{0}^{\pi} \int_{0}^{2\pi} \int_{0}^{5} \rho^{6} \sin \phi d \rho d \phi d \theta$

$\int_{0}^{\pi} \int_{0}^{2\pi}|_{0}^{5} \frac{1}{7} p^{7} \sin \phi d \phi d \theta$

$=\int_{0}^{2\pi} \left.\frac{\left(5)^{7}\right.}{7}(-\cos \phi)\right|_{0}^{\pi} d \theta$

$=\frac{(5)^{7}}{7} *(2 \pi) *[-\cos \pi+\cos 0]$

$=\frac{312500 \pi}{7}$