Question Number 129975 by Algoritm last updated on 21/Jan/21
Answered by Olaf last updated on 21/Jan/21
$$\Omega\:=\:\int_{−\infty} ^{+\infty} {e}^{−\left(\frac{{x}}{{a}}\right)^{\mathrm{2}} } {dx} \\ $$$$\Omega\:=\:\mathrm{2}\mid{a}\mid\int_{\mathrm{0}} ^{+\infty} {e}^{−{u}^{\mathrm{2}} } {du} \\ $$$$\Omega\:=\:\mathrm{2}\mid{a}\mid\frac{\sqrt{\pi}}{\mathrm{2}} \\ $$$$\Omega\:=\:\mid{a}\mid\sqrt{\pi} \\ $$
Answered by MJS_new last updated on 21/Jan/21
$$\underset{\mathbb{R}} {\int}\mathrm{e}^{−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}} {dx}=\mathrm{2}\underset{\mathbb{R}} {\int}\mathrm{e}^{−{x}^{\mathrm{2}} } {dx}=\mathrm{2}\sqrt{\pi} \\ $$