How to prove the value of this integral?

[suchith]

$\int^{∞}_{-∞} e^{-x^{2}}dx$ = $\frac{\sqrt{\pi}}{2}$

 

[pwsnafu]

Ah, that one. Hint: consider
$\int_{-\infty}^{\infty} \int_{-\infty}^\infty e^{-x^2-y^2}\, dx \, dy$

 

[HallsofIvy]

That should be found in pretty much any Calculus text. Look at the integral pwsnafu suggests. Note that, by symmetry, that is $4\int_0^\infty\int_0^\infty e^{-x^2- y^2} dx dy$, over the first quadrant. That can be converted into a "doable" integral by changing to polar coordinates.

 

[suchith]

That should be found in pretty much any Calculus text. Look at the integral pwsnafu suggests. Note that, by symmetry, that is $4\int_0^\infty\int_0^\infty e^{-x^2- y^2} dx dy$, over the first quadrant. That can be converted into a "doable" integral by changing to polar coordinates.

I don't know Double integrals. Is it possible to prove the result only using Single variable calculus? At first I tried Integration by parts, but I failed :(

Isn't it possible to integrate using limit of sums and symmetry with suitable manipulations? I tried to sum directly but failed :(