This applet is designed to let you explore arclength of the graph of a function.[br][br]A curve y=f(x) is given and compared to the polygonal path obtained by cutting the x interval in half a number of times and connecting points on the curve by straight lines.[br][br]ds is the normal integration factor evaluated at the midpoints of the straight lines.[br][math]\Delta s[/math] is the ratio of each straight line to its base.[br][br]If you cut the interval in half 10 times, there are over 1000 subintervals and the approximation is very good.

As always, when exploring with a new method, start with a curve whose length you know.[br][br][list][*]That is easy to do with a straight line where f(x)= ax.[/*][*]You can also look at a quarter circle.[br](If you try a half circle the derivative becomes undefined at the endpoints. You can however use endpoints that are very close to the points with vertical tangents.)[/*][*]You can then look at arbitrary functions.[/*][*]It is fun to look at a function that is vary curvy, like f(x)=sin(10x)[/*][/list][br]In each case it is worthwhile to:[br][list][*]See how many subdivisions you need to make for the polygonal length to be within 1, 2, or 3 decimal places of the integral answer.[/*][*]See how widely ds varies in the interval.[/*][/list]