( ∫_{−∞}^{∞} e^{−x^{2}} dx)^{2} = ∫_{−∞}^{∞} ∫_{−∞}^{∞} e^{−(x^{2}+y^{2})} dx dy

= ∫_{0}^{2π}∫_{0}^{∞}e^{−r^{2}}r dr dθ

= ∫_{0}^{2π}[−\frac{e^{−r^{2}}}{2} ]_{r=0}^{r=∞} dθ
= π.