Question

Evaluate (x2 + y2) dv e , where e lies between the spheres x2 + y2 + z2 = 9 and x2 + y2 + z2 = 16.

Answer 1

Convert to spherical coordinates, using

$x(\rho,\theta,\varphi)=\rho\cos\theta\sin\varphi$
$y(\rho,\theta,\varphi)=\rho\sin\theta\sin\varphi$
$z(\rho,\theta,\varphi)=\rho\cos\varphi$

The volume element is

$\mathrm dV=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi$

In spherical coordinates, the two given spheres are obtained by setting $\rho=3$ and $\rho=4$. So the integral evaluates to

$\displaystyle\iiint_{\mathcal E}(x^2+y^2)\,\mathrm dV=\int_{\varphi=0}^{\varphi=\pi}\int_{\theta=0}^{\theta=2\pi}\int_{\rho=3}^{\rho=4}\rho^2\sin^2\varphi(\rho^2\sin\varphi)\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi$
$=\dfrac{6248\pi}{15}$