Compute ∫_0^∞ e^{-x^2} dx.

Compute ∫_0^∞ e^{-x^2} dx.

Approach: Square the integral and convert to polar coordinates.

√π/2. Let I = ∫_0^∞ e^{-x^2} dx. Then I² = ∫_0^∞ ∫_0^∞ e^{-(x^2+y^2)} dx dy = ∫_0^{π/2} ∫_0^∞ e^{-r^2} r dr dθ = π/4. So I = √π/2.

Follow-up: What is ∫_0^∞ x^2 e^{-x^2} dx?

Key concepts: polar coordinates, gaussian integral, double integral.