Find the total length of $r=e^{- \theta}$ , $\theta \geq 0$

Using the arc length formula in polar coordinates, 
\[L=\int_0^{\infty}\sqrt{r^2+(\frac{dr}{d\theta})^2}  d\theta=\int_0^{\infty}\sqrt{e^{-2\theta}+e^{-2\theta}}d \theta=\int_0^{\infty}\sqrt{2}e^{-\theta}d \theta\]
\[=-\sqrt{2} e^{-\theta}|_0^{\infty}=0-(-\sqrt{2})=\sqrt{2}.\]