Solving: $\int_0^\infty{e^{-x^2}}dx$

$\begin{aligned} I & =\int_0^{\infty} e^{-x^2} d x \\ \Rightarrow I^2 & =\left(\int_0^{\infty} e^{-x^2} d x\right)^2=\left(\int_0^{\infty} e^{-x^2} d x\right)\left(\int_0^{\infty} e^{-y^2} d y\right) \\ & =\int_0^{\infty} \int_0^{\infty} e^{-\left(x^2+y^2\right)} d x d y=\int_0^{\infty} \int_0^\infty{e^{-\left(x^2+y^2\right)}} d A \\ & =\int_0^{\pi / 2} \int_0^{\infty} e^{-r^2} r d r d \theta \\ & =\frac{\pi}{4} \int_0^{\infty} 2 e^{-r^2} r d r=\frac{\pi}{4} \\ \Rightarrow I & =\sqrt{\frac{\pi}{4}}=\frac{\sqrt{\pi}}{2}\end{aligned}$