cowardly9030 cowardly9030
  • 22-07-2017
  • Mathematics
contestada

Evaluate the integral below, where e lies between the spheres x2 + y2 + z2 = 9 and x2 + y2 + z2 = 25 in the first octant.

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LammettHash
LammettHash LammettHash
  • 25-07-2017
Assuming you're computing the volume of the region [tex]E[/tex], we have upon converting to cylindrical coordinates,

[tex]\displaystyle\iiint_E\mathrm dV=\int_{\varphi=0}^{\varphi=\pi/2}\int_{\theta=0}^{\theta=\pi/2}\int_{\rho=3}^{\rho=5}\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi[/tex]
[tex]=\displaystyle\frac\pi2\left(\int_0^{\pi/2}\sin\varphi\,\mathrm d\varphi\right)\left(\int_3^5\rho^2\,\mathrm d\rho\right)[/tex]
[tex]=\dfrac{49\pi}3[/tex]
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