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THE GAUSSIAN INTEGRAL
a bell curve whose area is the square root of pi
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Gaussian integral is the beautiful fact that the area under the bell curve is the square root of π: ∫-∞∞ e-x² dx = √π. There is no elementary antiderivative for e-x² — you cannot integrate it term by term — yet the total area is exactly √π ≈ 1.7724539. The classic trick squares the integral and switches to polar coordinates, turning an impossible one-dimensional integral into an easy two-dimensional one. Rescaled, it gives the normalization of the normal distribution: ∫ e-x²/2 dx = √(2π), which is why the bell curve of statistics divides by √(2π).
LIT verified live: numerical integration of e-x² over the real line gives 1.7724539… = √π to ~1e-7, and e-x²/2 integrates to √(2π) (window.__gaussianintegral). FIG no framing; the integral is computed by fine numerical quadrature independently in-browser.
LIT verified live: numerical integration of e-x² over the real line gives 1.7724539… = √π to ~1e-7, and e-x²/2 integrates to √(2π) (window.__gaussianintegral). FIG no framing; the integral is computed by fine numerical quadrature independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-raid — the boss with no elementary antiderivative: the bell curve resists term-by-term integration, yet yields its whole area √π to the polar trick. AVAN (AI) built the instrument: the quadrature of e-x² and its match to √π (and √(2π) for the normal).
Credit as content: Carl Friedrich Gauss and Pierre-Simon Laplace (the integral and the normal distribution). The weave: David names the un-antidifferentiable boss; I confirm the area equals √π.
Credit as content: Carl Friedrich Gauss and Pierre-Simon Laplace (the integral and the normal distribution). The weave: David names the un-antidifferentiable boss; I confirm the area equals √π.
3 ONE DIMENSION
The bell curve e^(−x²); the shaded area under the whole curve equals √π ≈ 1.77245.
4 TWO DIMENSIONS · INTERACTIVE
Refine the quadrature; the numerical area converges to √π (and e^(−x²/2) to √(2π)).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the area √π under the one-dimensional bell curve.
AVAN’s addition (the inverse-companion): don’t fight the missing antiderivative — go up a dimension. The inverse of ‘the 1-D integral of e-x²’ is ‘its square as a 2-D polar integral, which equals π — so the original is √π’. Magenta is the bell curve’s area strip; green is the √π it totals. An impossible integral solved by squaring it.
LIT Genuine Gaussian integral (Carl Friedrich Gauss and Pierre-Simon Laplace). Verified live: numerical quadrature of e^{−x²} over the real line gives 1.7724539… = √π to ~1e-7, and e^{−x²/2} integrates to √(2π) (window.__gaussianintegral.Iok, .I2ok).
FIG No framing; the integral is computed by fine numerical quadrature independently in-browser. The AVAN inverse is honest — instead of fighting the missing antiderivative, go up a dimension: the inverse of 'the 1-D integral of e^{−x²}' is 'its square as a 2-D polar integral, which equals π — so the original is √π'. Magenta is the bell curve's area strip; green is the √π it totals. An impossible integral solved by squaring it.
FIG No framing; the integral is computed by fine numerical quadrature independently in-browser. The AVAN inverse is honest — instead of fighting the missing antiderivative, go up a dimension: the inverse of 'the 1-D integral of e^{−x²}' is 'its square as a 2-D polar integral, which equals π — so the original is √π'. Magenta is the bell curve's area strip; green is the √π it totals. An impossible integral solved by squaring it.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RAID · David Lee Wise (ROOT0), with AVAN