Question

Use spherical coordinates. evaluate (x2 + y2) dv e , where e lies between the spheres x2 + y2 + z2 = 1 and x2 + y2 + z2 = 25.

Answer 1

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

$x^2+y^2=\rho^2\cos^2\theta\sin^2\varphi+\rho^2\sin^2\theta\sin^2\varphi=\rho^2\sin^2\varphi$

$\displaystyle\iiint_E(x^2+y^2)\,\mathrm dV=\int_{\varphi=0}^{\varphi=\pi}\int_{\theta=0}^{\theta=2\pi}\int_{\rho=1}^{\rho=5}\rho^4\sin^3\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi$
$=\displaystyle2\pi\left(\int_{\varphi=0}^{\varphi=\pi}\sin^3\varphi\,\mathrm d\varphi\right)\left(\int_{\rho=1}^{\rho=5}\rho^4\,\mathrm d\rho\right)$
$=\dfrac{24992\pi}{15}$