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^(√(1-x^2))\int_0^(√(1-x^2-y^2)) xe^((x^2+y^2+z^2))dzdydx=0.392699`