Problem 1: Evaluate $I = \int \int \int_E e^{{(x^2+y^2+z^2)}^{\frac{3}{2}}} dV $ , where $E$ is the portion of the unit ball $x^2 + y^2 + z^2 \leq 1$ that lies in the first octant (i.e. the region with $x \geq 0, y \geq 0, z \geq 0$).
I tried the solution of where $x^2+y^2+z^2=\rho^2$ and substitute this and get this integral in cylindrical coordinates
$$\int_{0}^1 \int_{0}^\frac{\pi}{2} \int_{0}^\frac{\pi}{2} e^{{(\rho^2)}^{\frac{3}{2}}}d \rho d\theta d\phi $$ This is the substituion I made for $x^2+y^2+z^2$ and transformed this into cylindrical coordinates. I tried evaluating $d\rho$ but it uses a gamma function on wolfram. Can anyone tell me how to take this further? Thank you.
Your transformation is wrong. Note that $dV$ isn't $d\rho\;d\theta\;d\phi$, it is $\rho^2\;d\rho\;d\theta\;d\phi$.
This, together with the fact that $(\rho^2)^{3/2}=\rho^3$, should make the problem simple.