Calculate $\int_0 ^\infty\int_0 ^\infty e^{-x^2 -y^2} dx dy$

Question

I want to calculate the following double integral:

$$\int_{0}^{\infty}\int_{0}^{\infty} e^{-x^2 -y^2} \ \mathrm{d}x \ \mathrm{d}y$$

I used the change of variable $x=r\cos\theta$, $y=r\sinθ$, so I get

$$\int_{0}^{\infty}\int_{0}^{\infty} e^{-r^2}r \ \mathrm{d}r \ \mathrm{d}\theta$$ $$\frac{-1}{2}\int_{0}^{\infty} e^{-r^2} \ \mathrm{d}\theta$$

I'm stuck here. How can I calculate that integral?

Thanks.

Answer 1

For $$\int_0^\infty e^{-r^2}r\,{\rm d}r$$

Consider the substitution $u=r^2$

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For the record, your integral should be

$$\int_0^{\pi/2}\int_0^\infty e^{-r^2}r\,{\rm d}r\,{\rm d}\theta$$