Question

Evaluate xe^(x^2+y^2+z^2) where e is the portion of the unit ball x^2+y^2+z^2 < 1 that lies in the firat octant

Answer 1

Given

$\int\int\int_E xe^{(x^2+y^2+z^2)}$

where E is the portion of the unit ball $x^2+y^2+z^2\leq1$ that lies in the first octant.

This can be evaluated as follows:

$\int_0^1\int_0^{\sqrt{1-x^2}}\int_0^{\sqrt{1-x^2-y^2}} xe^{(x^2+y^2+z^2)}dzdydx=0.392699$