Gaussian Integral

Tap a term to see what it means, or .

Calculate

Example: the area under the curve y = e^(−x²) over the whole line is exactly √π ≈ 1.772454. Adding up 12 rectangles between −3 and 3, how close does it get? Change the values to work out your own case, including the a in e^(−ax²).

a in e^−ax² Limit L (from −L to L) Rectangles n

Step by step

What each term means

Integral over the Whole Line: The area under the curve from −∞ to +∞. The curve falls so fast that almost all the area is near zero: between −3 and 3 lies 99.998% of it.
Gaussian Curve (e^(−x²)): The bell without scaling. It has no antiderivative written with elementary functions, so the area doesn't come out of the usual integration rules.
Square Root of Pi (√π): The exact value, which Gauss got by squaring the integral: it becomes a double integral over the plane, which polar coordinates solve. This is where the √(2π) of the normal distribution comes from.