How to compute the integral $\int_{-\infty}^\infty e^{-x^2}\,dx$?

Question

How to compute the integral $\int_{-\infty}^\infty e^{-x^2}\,dx$ using polar coordinates?

Answer 1

Hint: Let $I=\int_{-\infty}^\infty e^{-x^2}\,dx.$ Then $$I^2=\left(\int_{-\infty}^\infty e^{-x^2}\,dx\right)\left(\int_{-\infty}^\infty e^{-y^2}\,dy\right)=\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-x^2}e^{-y^2}\,dx\,dy=\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-(x^2+y^2)}\,dx\,dy.$$ Now switch to polar coordinates.

Answer 2

The usual tear-free approach is to write $$\left(\int_{-\infty}^\infty e^{-x^2}\,dx\right)^2=\int_{-\infty}^\infty e^{-x^2}\,dx\int_{-\infty}^\infty e^{-y^2}\,dy=\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-(x^2+y^2)}\,dx\,dy$$ and change to polar coordinates.