One hard integral becomes one easy radius.
Squaring creates a radially symmetric surface. Every direction becomes equivalent, leaving only distance from the origin.
Analytic proof
Equation Lab / Analysis & probability
Square the curve.
Reveal π.
The Gaussian integral becomes solvable when one dimension becomes two. Move through the proof and watch symmetry turn an impossible elementary antiderivative into a radial volume.
03 / From proof to data
Normalize real uncertainty.
The standard normal density is not e⁻ˣ². It is φ(x) = (1/√2π)e⁻ˣ²⁄². That normalization makes the total probability exactly one.
Simulated measurement residualsφ(x) theoretical curve
The extra r counts expanding rings.
A polar cell has area r dr dθ. Without the Jacobian, outer rings would be undercounted and the volume would be wrong.
Normalization turns shape into probability.
The Gaussian’s exact area supplies the constant behind noise models, uncertainty propagation, likelihoods, and experimental error bars.