Verified answer

Using spherical coordinates (y ≥ 0 ==> θ = 0 to π), this equals

∫(θ = 0 to π) ∫(φ = 0 to π) ∫(ρ = 0 to 2) e^ρ * (ρ^2 sin φ dρ dφ dθ)

= π ∫(φ = 0 to π) sin φ dφ * ∫(ρ = 0 to 2) ρ^2 e^ρ dρ

= 2π * (ρ^2 - 2ρ + 2) e^ρ {for ρ = 0 to 2}

= 2π(2e^2 - 2)

= 4π(e^2 - 1).

I hope this helps!

Switch to spherical polar coordinates such that the integral is of e^(r^2) over the volume (rsin(theta))^2drd(theta)d(phi), and make sure you rewrite the limits as well in spherical polar coordinates using your standard coordinate transformations. Good luck