as part of a math tutorial I am supposed to evaluate the following integral:
$$ \int_{-1}^1 \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} \int_{-\sqrt{1-x^2-y^2}}^\sqrt{1-x^2-y^2} e^{(x^2+y^2+z^2)^\frac{3}{2}} dz \ dy \ dx$$
Now I am pretty sure this is over the unit sphere with $$-1\leq x \leq 1, \ \ \ -\sqrt{1-x^2}\leq y \leq \sqrt{1-x^2}, \ \ \ -\sqrt{1-x^2-y^2}\leq z \leq \sqrt{1-x^2-y^2}\ \ \ $$
so I tried using spherical coordinates and changing the integral to
$$ \int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^1 e^{(\rho^2)^\frac{3}{2}}\rho^2 \sin \phi \ d\rho \ d\phi \ d \theta$$ which simplifies to $$ \int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^1 e^{\rho^3}\rho^2 \sin \phi \ d\rho \ d\phi \ d \theta$$
Already in this step, I am unsure if the new integral boundaries are correct and if the substitution is correct. I tried plugging in $\rho = 1$, but this does not yield the correct result. (WolframAlpha says the value should be ~7.19)
The integral looks fine. Using the $u$-substitution you suggested in the comments, you should get $4\pi(e-1)/3$, which agrees with WolframAlpha.