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nalin [4]
4 years ago
9

Se spherical coordinates to evaluate the triple integral ∭e(x2+y2+z2)dv, where e is the ball: x2+y2+z2≤81.

Mathematics
1 answer:
meriva4 years ago
8 0
Converting to spherical coordinates, we use

x=\rho\cos\theta\sin\varphi
y=\rho\sin\theta\sin\varphi
z=\rho\cos\varphi
\implies\mathrm dV=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi

Then

\displaystyle\iiint_E(x^2+y^2+z^2)\,\mathrm dV=\int_{\varphi=0}^{\varphi=\pi}\int_{\theta=0}^{\theta=2\pi}\int_{\rho=0}^{\rho=9}\rho^4\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=\frac{236,196\pi}5
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