ntm

Squaring the impossible integral unlocks the circle. The antiderivative of e to the −x² does not exist in elementary functions, so the definite integral from −∞ to ∞ cannot be found directly. Squaring it produces a double integral of e to the −(x² + y²) over the entire plane. Polar coordinates make the exponent depend only on radius. The area element r dr dθ lets the angle integrate to 2π while the radial integral, after u = r², collapses to 1, giving exactly π. The original integral is therefore √π. An infinitely wide bell curve still encloses a finite area fixed by the circle.