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

Convert to spherical coordinates, using

[tex]x(\rho,\theta,\varphi)=\rho\cos\theta\sin\varphi[/tex]
[tex]y(\rho,\theta,\varphi)=\rho\sin\theta\sin\varphi[/tex]
[tex]z(\rho,\theta,\varphi)=\rho\cos\varphi[/tex]

The volume element is

[tex]\mathrm dV=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi[/tex]

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

[tex]\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[/tex]
[tex]=\dfrac{6248\pi}{15}[/tex]