Question

Evaluate the integral below by changing to spherical coordinates.

a) ∫∫∫ e^(-x² - y² - z²) dV
b) ∫∫ e^(-x² - y²) dA
c) ∫ e^(-r²) r² sin(θ) dr dθ dϕ
d) ∫∫ e^(-r²) r dr dθ

Answer 1

To evaluate the given integrals using spherical coordinates, we make appropriate substitutions and integrate over the spherical coordinates.

Step-by-step explanation:

To evaluate the given integrals using spherical coordinates, we need to make the following substitutions:

a) In the integral ∫∫∫ e^(-x² - y² - z²) dV, we substitute x = ρsin(φ)cos(θ), y = ρsin(φ)sin(θ), and z = ρcos(φ) to obtain the integral ∫∫∫ e^(-ρ²)ρ² sin(φ) dρ dφ dθ.

b) In the integral ∫∫ e^(-x² - y²) dA, we substitute x = rcos(θ) and y = rsin(θ) to obtain the integral ∫∫ e^(-r²) r dr dθ.

c) In the integral ∫ e^(-r²) r² sin(θ) dr dθ dϕ, we directly integrate over the given spherical coordinates.