always remember that:
I = ∫ − ∞ ∞ e − x 2 d x {\displaystyle I=\int _{-\infty }^{\infty }e^{-x^{2}}dx}
I 2 = ( ∫ − ∞ ∞ e − x 2 d x ) ⋅ ( ∫ − ∞ ∞ e − y 2 d y ) {\displaystyle I^{2}=\left(\int _{-\infty }^{\infty }e^{-x^{2}}dx\right)\cdot \left(\int _{-\infty }^{\infty }e^{-y^{2}}dy\right)}
I 2 = ∬ R 2 e − ( x 2 + y 2 ) d ( x , y ) {\displaystyle I^{2}=\iint _{\mathbb {R} ^{2}}e^{-(x^{2}+y^{2})}d(x,y)}
I 2 = ∫ 0 2 π ∫ 0 ∞ e − r 2 | ∂ x ∂ r ∂ x ∂ θ ∂ y ∂ r ∂ y ∂ θ | d r d θ {\displaystyle I^{2}=\int _{0}^{2\pi }\int _{0}^{\infty }e^{-r^{2}}{\begin{vmatrix}{\frac {\partial x}{\partial r}}&{\frac {\partial x}{\partial \theta }}\\{\frac {\partial y}{\partial r}}&{\frac {\partial y}{\partial \theta }}\\\end{vmatrix}}\ dr\,d\theta }
I 2 = 2 π ∫ 0 − ∞ e − r 2 | cos ( θ ) − r sin ( θ ) sin ( θ ) r cos ( θ ) | d r {\displaystyle I^{2}=2\pi \int _{0}^{-\infty }e^{-r^{2}}{\begin{vmatrix}\cos(\theta )&-r\sin(\theta )\\\sin(\theta )&r\cos(\theta )\\\end{vmatrix}}\,dr}
I 2 = π ∫ 0 ∞ 2 r e − r 2 d r {\displaystyle I^{2}=\pi \int _{0}^{\infty }2re^{-r^{2}}dr}
I 2 = π ⋅ 1 {\displaystyle I^{2}=\pi \cdot 1}
I 2 = π {\displaystyle I^{2}=\pi }
I = π {\displaystyle I={\sqrt {\pi }}}
pyrolysis oven svg
png
