Question

Se spherical coordinates to evaluate the triple integral ∭e(x2+y2+z2)dv, where e is the ball: x2+y2+z2≤81.

Answer 1

Converting to spherical coordinates, we use

$x=\rho\cos\theta\sin\varphi$
$y=\rho\sin\theta\sin\varphi$
$z=\rho\cos\varphi$
$\implies\mathrm dV=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi$

Then

$\displaystyle\iiint_E(x^2+y^2+z^2)\,\mathrm dV=\int_{\varphi=0}^{\varphi=\pi}\int_{\theta=0}^{\theta=2\pi}\int_{\rho=0}^{\rho=9}\rho^4\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=\frac{236,196\pi}5$