Evaluating $\int_{0}^{\infty} \int_{0}^{\infty} xe^{-(x^2+y^2)} dx dy$

Asked Mar 22 '17 at 23:35

Active Mar 22 '17 at 23:44

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This is the integral: $$\int_{0}^{\infty} \int_{0}^{\infty} xe^{-(x^2+y^2)} dx dy$$ So i have this: $$\int_{0}^{\infty}e^{-y^2}\left( \int_{0}^{\infty}xe^{-x^2}dx\right)dy= \int_{0}^{\infty} -\frac{e^{-y^2}}{2} \left(\lim_{b \to \infty} e^{-b}-e^0\right)$$ $$=\frac{1}{2}\int_{0}^{\infty}e^{-y^2}dy$$

How can i resolve that last integral. And i know that the answer is $\frac{\sqrt \pi}{4}$

edited Mar 22 '17 at 23:44

Olivier Oloa

120,989

asked Mar 22 '17 at 23:35

Lexie Walker

2

Here you have many ways to conclude:http://math.stackexchange.com/questions/9286/proving-int-0-infty-mathrme-x2-dx-dfrac-sqrt-pi2 – Olivier Oloa Mar 22 '17 at 23:40
You are welcome. – Olivier Oloa Mar 22 '17 at 23:43
1

This question has been solved by the comment of @OlivierOloa. Is there a way to mark it as answered? – toliveira Mar 22 '17 at 23:45
Should be marked as duplicate then. – Zaid Alyafeai Mar 23 '17 at 03:06

Evaluating $\int_{0}^{\infty} \int_{0}^{\infty} xe^{-(x^2+y^2)} dx dy$

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