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Use spherical coordinates to evaluate the triple integral

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where E is the region bounded by the spheres enter image description here and enter image description here

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Transform to spherical coordinates to obtain the integral $$ I_E = \int_2 ^3 \int_0 ^ {2\pi} \int_0 ^\pi \frac{e^{-\rho^2}}{\rho} \rho^2 \sin \phi \, \mathrm{d} \phi \, \mathrm{d} \theta \, \mathrm{d} \rho = \cdots $$ To evaluate the integral, you can employ Fubini's Theorem to get the value quite handily. I believe the result is $I_E = \frac{2 \pi}{e^4} \left( 1 - \frac{1}{e^5} \right)$.

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  • $\begingroup$ @user131040 No problem! I am glad that it was helpful - when just learning multivariable, the different coordinate maps can seem more foreboding at first, I am happy that this was useful for you! $\endgroup$ – izœc Mar 27 '14 at 6:06

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