Set up the integral that represents the arc length of the curve f(x) = ln(x) + 5 on [1, 3], and then use simpson's rule with n = 4 to approximate the arc length. then compare the approximation to the actual arc length found by integrating with technology.

The integral for the arc of length is:

By using Simpon’s rule we get: 1.5355453

And using technology we get:  2.3020

The approximation is about 33% smaller than the exact result.

Explanation:

The formula for the length of arc of the function f(x) in the interval [a,b] is:

We need the derivative of the function:

And we need it squared:

Then the integral is:

Now, the Simposn’s rule with n=4 is:

In this problem:

So, the Simposn’s rule formula becomes:

Then simplifying a bit:

Then we just do those computations and we finally get the approximation via Simposn's rule:

While when we do the integral by using technology we get: 2.3020.

The approximation with Simpon’s rule is close but about 33% smaller: