Finding the Arc Length of a Logarithmic Spiral

Find the arc length \(s\) of the logarithmic spiral represented by \(r=f(\theta)=e^{3\theta }\) from \(\theta = 0\) to \(\theta =2.\)

Solution We use the arc length formula \(s=\int_{\alpha }^{\beta } \sqrt{\,r^{2}+\left( \dfrac{dr}{d\theta }\right) ^{2}}\,d\theta\) with \(r=e^{3\theta }\). Then \(\dfrac{dr}{d\theta }=3e^{3\theta }\) and \begin{eqnarray*} s&=&\int_{0}^{2}\sqrt{(e^{3\theta })^{2}+(3e^{3\theta })^{2}}\,d\theta =\int_{0}^{2}\sqrt{10e^{6\theta }}d\theta =\sqrt{10}\int_{0}^{2}e^{3\theta }\,d\theta\\ &=& \sqrt{10}\left[\dfrac{e^{3\theta}}{3}\right]^2_0 =\frac{\sqrt{10}}{3}(e^{6}-1) \end{eqnarray*}