Gaussian integral by switching to polar coordinates

Question

Use polar coordinates to compute $I=\displaystyle\int_{-\infty}^{\infty}e^{-x^2/2}\,dx$ .

Short answer

$\sqrt{2\pi}$ . Square it: $I^2=\iint e^{-(x^2+y^2)/2}\,dx\,dy=\int_0^{2\pi}\!\int_0^\infty e^{-r^2/2}r\,dr\,d\theta=2\pi\cdot1$ . Since $I>0$ , $I=\sqrt{2\pi}$ .

Adapted from Zhou, A Practical Guide to Quantitative Finance Interviews, sec. 3.3 Partial Derivatives and Multiple Integrals, 'Gaussian integral', p.40; question and answer rewritten by SelfMadeQuant.com.

See the hint and the full worked solution in the question bank (free account).

Open the worked solution

Lessons behind this question

Integration Strategy
The Normal Distribution
The Fundamental Theorem of Calculus

Start with a free lesson: Normal Distribution is part of a free topic. Create a free account, no card needed, to read it in full with worked solutions.

Read it free