Arc Length — Question 8

Question 8 – Solution

Step 1: List the three points. $$(0,0),\qquad\left(\frac\pi2,1\right),\qquad(\pi,0).$$ Step 2: Find the first chord length. $$\begin{align*} d_1&=\sqrt{\left(\frac\pi2-0\right)^2+(1-0)^2}\\ &=\sqrt{\frac{\pi^2}{4}+1}. \end{align*}$$ By symmetry, $d_2=d_1$. Thus $$\begin{align*} L_{\mathrm{chord}}&=2d_1 =2\sqrt{\frac{\pi^2}{4}+1} =\sqrt{\pi^2+4}. \end{align*}$$ Step 3: Compare with the true length. The true arc length is $$L=\int_0^\pi\sqrt{1+\cos^2x}\,dx.$$ A curved arc between two points is longer than its straight chord. Since the sine curve is not straight on either half, $$\boxed{L_{\mathrm{chord}}=\sqrt{\pi^2+4},\qquad L>L_{\mathrm{chord}}}$$