Problem

Find the length of the curve y = sin(x) from x = 0 to x = p.

Solution

As a rough guess, we see the sine curve in red is not far from the straight lines connecting (0, 0) with (p/2, 1) and (p/2, 1) with (p, 0). So we can estimate the arclength to be a bit more than

Set up the integral:

As simple as this integral is to set up, there is no nice form for the antiderivative. The only thing we can do is estimate using a Riemann sum or trapezoids. Here is an estimate using the Numerical Integration applet and the trapezoid rule.

This is the values of the estimate, shown to 3 decimal places.