Problem

Find the arclength of the function

between the values x = 0 and x = 2. Give an estimate using the trapezoid rule with 4 intervals, then with 16 intervals.

Solution

Here's a sketch of the function, showing the length of arc to be estimated. Notice that the x and y scales are different

The Arclength Formula

The length of the curve y = f(x) from a to b is the value of the integral

In many cases, we cannot get an antiderivative and so we must do with an estimate. Let's set up the integral.

Given

First compute the derivative:

Next, square it:

Compute

and simplify.

So the integral to solve is

Using the trapezoid rule with 4 intervals

The trapezoid rule is

The length h of each subinterval is

and the yi's are the values of the integrand at the points

Calculate the value of

at each of the 5 points.

The trapezoid rule,

becomes

Since this is an estimate itself, combine the terms and express the answer as a decimal number.

How good is this estimate? Let's see. There is an antiderivate of

that we found this using a computer algebra program:

The exact arclength is

when we write it as a decimal to three places of accuracy.

Using the trapezoid rule with 16 intervals

To compute the estimate using 16 subintervals, h becomes

The y_i's are

and so on, up to

The trapezoid rule estimate becomes

Better this time.