Question

(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.

$\begin{cases}x=\rho\cos\theta\sin\varphi\\y=\rho\sin\theta\sin\varphi\\z=\rho\cos\varphi\end{cases}\implies\mathrm dV=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi$

$\displaystyle\iiint_E(x^2+y^2)\,\mathrm dV=\int_{\varphi=0}^{\varphi=\pi}\int_{\theta=0}^{\theta=2\pi}\int_{\rho=3}^{\rho=4}\underbrace{\rho^2\sin^2\varphi}_{x^2+y^2}\underbrace{\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi}_{\mathrm dV}$
$=\displaystyle2\pi\left(\int_{\varphi=0}^{\varphi=\pi}\sin^3\varphi\,\mathrm d\varphi\right)\left(\int_{\rho=3}^{\rho=4}\rho^4\,\mathrm d\rho\right)=\frac{6248\pi}{15}$