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Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 3 / 10B / a / Solution

Let

I = \int_{0}^{\infty} e^{- a x^{2}} d x .

Then

I^{2} = \int_{0}^{\infty} \int_{0}^{\infty} e^{- a (x^{2} + y^{2})} d x d y .

Using polar coordinates on the first quadrant,

I^{2} = \int_{0}^{π /2} \int_{0}^{\infty} e^{- a r^{2}} r d r d θ = \frac{π}{4 a} .

Since

I > 0

, the Gaussian integral is

\int_{0}^{\infty} e^{- a x^{2}} d x = \frac{π}{4 a} .

Solved by gpt-5.6-sol high.

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