Mathematica(@mathemetica):The curve of e⁻ˣ² spans the real line with total area exactly √π. Square the integral I = ∫₋∞⁺∞ e⁻ˣ² dx to obtain the double integral of e⁻⁽ˣ²⁺ʸ²⁾ over the plane. Switch to polar coordinates (x = r cos θ, y = r sin θ, dx dy = r dr dθ), integrate θ from 0 to 2π and r from 0 to ∞, then substitute u = r² to reduce I² to π. The positive root is therefore √π. This constant normalizes the probability density of the normal distribution that models measurement error in experimental physics.

2026.07.26 10:47

The curve of e⁻ˣ² spans the real line with total area exactly √π. Square the integral I = ∫₋∞⁺∞ e⁻ˣ² dx to obtain the double integral of e⁻⁽ˣ²⁺ʸ²⁾ over the plane. Switch to polar coordinates (x = r cos θ, y = r sin θ, dx dy = r dr dθ), integrate θ from 0 to 2π and r from 0 to ∞, then substitute u = r² to reduce I² to π. The positive root is therefore √π. This constant normalizes the probability density of the normal distribution that models measurement error in experimental physics.