Gaussian Integration

Master the mathematics of the Gaussian Bell Curve. The standard integral, $\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$, forms the backbone of statistics, normal distributions, and quantum mechanics.

Interactive Sandbox

1D Bell Curve & 3D Volume Dome

Adjust amplitude $c$, width scale $a$, and center shift $b$ sliders. Drag directly on the 3D canvas to rotate the volume of revolution ($z = e^{-(x^2+y^2)}$) illustrating the polar coordinate transform.

1D Bell Curve: y = c * e^(-a(x-b)²)

3D Gaussian Dome (Drag to Rotate)

Gaussian Integral General Formula

∫_-∞^∞ c • e^-a(x-b)² dx = c • √(π / a)

Amplitude (c) 0.40

Width Scale (a) 0.50

Center Shift (b) 0.00

Observe how shifting the center (b) moves the curve but leaves the total integrated area completely unchanged.

Algebraic Solver

Polar Coordinate Proof Solver

Trace the double-integral polar transformation used to prove the Gaussian identity.

The Polar Coordinate Substitution Proof

Analytical Area Evaluation

Integral Term Coefficient

c = 0.40

Gaussian Area Result (∫ y dx)

Area = 0.40 * √(π / 0.50) = 1.0027

Volume of revolution (∫∫ z dA)

Volume = Area² = 1.0053

Evaluation

Theorem Quiz

Test your understanding of normal distributions, standard Gaussian limits, and Jacobians.

Question 1 of 5 Score: 0/0

What is the exact analytical value of ∫_-∞^∞ e^(-x²) dx?

Correct Answer!

Explanation text...

Gaussian Integrals Rules

Standard Integral: ∫ e^(-x²) dx = √π
Jacobian Polar: dx dy = r dr dθ
Normal Distribution: ∫ [1/√(2πσ²)] e^(-x²/2σ²) dx = 1