題目 4. (10 points) Find the arc length of the polar curve r=θr=\thetar=θ with 0≤θ≤π0 \le \theta \le \pi0≤θ≤π. 解答 解法一 思路 展開 極座標弧長公式為 L=∫r2+(dr/dθ)2 dθL=\int\sqrt{r^2+(\mathrm{d}r/\mathrm{d}\theta)^2}\,\mathrm{d}\thetaL=∫r2+(dr/dθ)2dθ。 本題 r=θr=\thetar=θ，積分可由標準的 ∫1+θ2 dθ\int\sqrt{1+\theta^2}\,\mathrm{d}\theta∫1+θ2dθ 公式完成。 答題過程 L= ∫0πθ2+1 dθ= 12[θθ2+1+ln(θ+θ2+1)]0π= 12(ππ2+1+ln(π+π2+1)).\begin{align*} L =&\,\int_0^\pi\sqrt{\theta^2+1}\,\mathrm{d}\theta\\[4mm] =&\,\frac12\left[\theta\sqrt{\theta^2+1} +\ln\big(\theta+\sqrt{\theta^2+1}\big)\right]_0^\pi\\[4mm] =&\,\boxed{\frac12\left(\pi\sqrt{\pi^2+1}+\ln\big(\pi+\sqrt{\pi^2+1}\big)\right)}. \end{align*}L===∫0πθ2+1dθ21[θθ2+1+ln(θ+θ2+1)]0π21(ππ2+1+ln(π+π2+1)).
