We are now ready to try to find the length of a curve defined by $y=f(x)$ for $a \leq x \leq b$, where the curve is not necessarily a straight line. Consider, for example, the graph $y=\sin x$ for $0 \leq x \leq \pi$, shown in Figure 4 .
a. A crude approximation to the length of the curve would be the straight line distance between the endpoints of the curve. Compute this distance. Is this value bigger or smaller than the actual arc length of this graph?
b. A better approximation would be found by dividing $[0, \pi]$ into two pieces and adding the straight line distance from $(0,0)$ to $\left(\frac{\pi}{2}, 1\right)$ to the distance from $\left(\frac{\pi}{2}, 1\right)$ to $(\pi, 0)$. See Figure 4. Compute this distance. How is it related to the answer in part a? How is it related to the actual arc length?
To get a more accurate answer, we will subdivide $[0, \pi]$ more finely yet. Copy the graph of $y=\sin x$ from Figure 4 into your notebook and sketch the straight line approximation to the curve using four subintervals of equal size. On another copy of this curve, sketch the straight line approximation to the curve using eight subintervals of equal size. In the lab, you will use the computer to calculate the lengths of these approximations.