I'm not sure whether this falls in the homework category, or the standard calculus section, so apologies in advance if this doesn't fall in the right category. 1. The problem statement, all variables and given/known data Evaluate ∫∫∫e^[(x^2 + y^2 + z^2)^3/2]dV, where the region is the unit ball x^2 + y^2 + z^2 ≤ 1. (or any variant of this question, where the region is always a ball with a radius of any size) 2. Relevant equations Relevant equations would be the conversion of rectangular coordinates to spherical coordinates, such as ρ^2 = x^2 + y^2 + z^2, as well as 3. The attempt at a solution Here, since the region is a whole sphere with a radius of one, I set the ranges for ρ to be from 0 to 1, and initially set the ranges for both angles from 0 to 2∏, and then set up a triple integral while substituting to get e^(r^3)*ρ^2sin∅dρdθd∅. However, I found out that the range for ∅ should be from 0 to ∏, instead of 2∏. Would it be possible to request an explanation as to why ∅ should only be what is essentially half a circle, while the other angle is 2∏? Thank you in advance.