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The Gaussian Integral.

The Math Flow

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15,157 次观看 • 4 个月前 •via X (Twitter)

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We often call Euler’s identity the most beautiful equation in mathematics. But sometimes we forget the beauty of the Gaussian integral. It is absolutely gorgeous. And if you don’t know this integral yet, can you really call yourself a math nerd? 😉

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$The Gaussian integral evaluates to √π: ∫_{−∞}^{∞} e^{−x²} dx = √π. The shaded region under y = e^{−x²} has this exact area. Letting I denote the integral, I² equals the volume of the surface z = e^{−(x² + y²)}, a rotationally symmetric hill whose volume is π when computed in polar coordinates.$

The Gaussian integral evaluates to √π: ∫_{−∞}^{∞} e^{−x²} dx = √π. The shaded region under y = e^{−x²} has this exact area. Letting I denote the integral, I² equals the volume of the surface z = e^{−(x² + y²)}, a rotationally symmetric hill whose volume is π when computed in polar coordinates.

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