# spherical coordinates to find the triple integral

Use spherical coordinates to evaluate the triple integral

where E is the region bounded by the spheres and

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 $I_E = \frac{2 \pi}{e^4} \left( 1 - \frac{1}{e^5} \right)$.