題目 Problem 3 (10%). Find the arc length of the polar curve: r=7θ,0≤θ≤2π.r=7^{\theta},\quad 0\le \theta\le 2\pi.r=7θ,0≤θ≤2π. 解答 解法一 極座標弧長公式為 L=∫02πr2+(drdθ)2 dθ.L=\int_0^{2\pi}\sqrt{r^2+\left(\frac{\mathrm{d}r}{\mathrm{d}\theta}\right)^2}\,\mathrm{d}\theta.L=∫02πr2+(dθdr)2dθ. 因為 r=7θr=7^\thetar=7θ、r′=(ln7)7θr'=(\ln7)7^\thetar′=(ln7)7θ，故 L=1+(ln7)2∫02π7θ dθ=1+(ln7)2ln7(72π−1).L=\sqrt{1+(\ln7)^2}\int_0^{2\pi}7^\theta\,\mathrm{d}\theta =\boxed{\frac{\sqrt{1+(\ln7)^2}}{\ln7}\left(7^{2\pi}-1\right)}.L=1+(ln7)2∫02π7θdθ=ln71+(ln7)2(72π−1).
