Using spherical coordinates, evaluate ∫ ∫ ∫v x*e^(x^2+y^2+z^2)^2 dV

Using spherical coordinates, evaluate ∫ ∫ ∫v x*e^(x^2+y^2+z^2)^2 dV, where V is the
solid that lies between the spheres x2 + y2 + z2 = 9 and x2 + y2 + z2 = 16 in the rst
octant.

Note that the spheres now have equations ρ = 3 and ρ = 4, respectively, in spherical coordinates.
(Also, both θ and φ are in [0, π/2], since we are in the first octant.)

Therefore,
∫∫∫ x e^(x^2+y^2+z^2)^2 dV
= ∫(θ = 0 to π/2) ∫(φ = 0 to π/2) ∫(ρ = 3 to 4) (ρ cos θ sin φ) e^(ρ^2)^2 * (ρ^2 sin φ dρ dφ dθ).

Evaluating this yields
∫(θ = 0 to π/2) cos θ dθ * ∫(φ = 0 to π/2) sin^2(φ) dφ * ∫(ρ = 3 to 4) ρ^3 e^(ρ^4) dρ
= ∫(θ = 0 to π/2) cos θ dθ * ∫(φ = 0 to π/2) (1/2)(1 - cos(2φ)) dφ * ∫(ρ = 3 to 4) ρ^3 e^(ρ^4) dρ
= [sin θ {for θ = 0 to π/2}] * [(1/2)(φ - sin(2φ)/2) {for φ = 0 to π/2}] * [(1/4) e^(ρ^4) {for ρ = 3 to 4}]
= 1 * π/4 * (1/4) (e^256 - e^81)
= (π/16) (e^256 - e^81).

I hope this helps!