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son4ous [18]
3 years ago
14

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

Mathematics
1 answer:
lara31 [8.8K]3 years ago
7 0
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}
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