Saturday, November 21, 2015

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




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


Answer



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.


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