Gaussian Integral
Area under the bell curve.
Key idea
Area under the bell curve.
In depth
∫₋∞^∞ e^(−x²) dx = √π. The trick is to square it, convert to a double integral, and switch to polar coordinates. It is the basis of the normal distribution.
Worked example
∫ e^(−x²/2) dx = √(2π). Hence the factor 1/√(2π) in the normal density.
Formula
Real-world uses
- Normalising the bell curve in probability and deriving formulas in quantum mechanics
Common mistake
Looking for an elementary antiderivative; there is none.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Gaussian Integral?
Area under the bell curve. It is a topic at the expert level in Calculus.
What is the formula for Gaussian Integral?
The key formula is ∫ e^(−x²)dx = √π.
Where is Gaussian Integral used in real life?
Real-world examples: Normalising the bell curve in probability and deriving formulas in quantum mechanics.
What should I learn before Gaussian Integral?
Make sure you are comfortable with Beta Function, the previous topic in Calculus.
What is a common mistake with Gaussian Integral?
Looking for an elementary antiderivative; there is none.
How can I practise Gaussian Integral?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Gaussian Integral?
Move on to Calculus of Variations Intro next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.