Exercise 2183Calculus › Multivariable calculus › Multiple integralsM1Added 3 Nov 2026

Gaussian integral over $\mathbb{R}^2$

#Gaussian#polar#improper integral

Statement

Show that $\displaystyle\iint_{\mathbb{R}^2}e^{-(x^2+y^2)}\,dx\,dy=\pi$ .

Hint

Polar coordinates: $r^2=x^2+y^2$ , Jacobian $r$ .

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Polar. $\int_0^{2\pi}\int_0^{+\infty}e^{-r^2}r\,dr\,d\theta=2\pi\cdot\frac12=\pi$ .

Conclusion. $\iint_{\mathbb{R}^2}e^{-(x^2+y^2)}\,dx\,dy=\pi$ .

Corollary. $\int_{-\infty}^{+\infty}e^{-x^2}\,dx=\sqrt\pi$ (Fubini argument).