In an exercise we had to calculate $\int\limits_{-\infty}^{+\infty}\int\limits_{-\infty}^{+\infty}e^{-(x_1^2+x_2^2)}dx_1dx_2$ and deduce $\int\limits_{-\infty}^{+\infty}e^{-x^2}dx$
To do this we calculate the double integral using polar coordinates, which gives $\pi$.
The second one is calculated like this:
$\int\limits_{-\infty}^{+\infty}e^{-x^2}dx=\sqrt{\int\limits_{-\infty}^{+\infty}e^{-x_1^2}dx_1\int\limits_{-\infty}^{+\infty}e^{-x_2^2}dx_2}=\sqrt{\int\limits_{-\infty}^{+\infty}e^{-(x_1^2+x_2^2)}dx_1dx_2}=\sqrt{\pi}$
Why were we allowed to merge those two integrals?
Best Answer
Because:$$\int\int f(x)g(y)\;dy\;dx=\int f(x)\int g(y)\;dy\;dx=\int g(y)\;dy\times \int f(x)\;dx$$
Here $f(x)=g(x)=e^{-x^2}$.