Trapezoid Rule and Simpson’s Rule

Consider the following integrals and the given values of n.

a. Find the Trapezoid Rule approximation to the integral using n and 2n subintervals.

b. Find the Simpson’s Rule approximation to the integral using 2n subintervals. It is easiest to obtain Simpson’s Rule approximations from the Trapezoid Rule approximations, as in Example 6.

c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.

Problem 32E

Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.

a. Find the Trapezoid Rule approximation to the integral using n and 2n subintervals.

b. Find the Simpson’s Rule approximation to the integral using 2n subintervals. It is easiest to obtain Simpson’s Rule approximations from the Trapezoid Rule approximations, as in Example 6.

c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.

Step 1:

Given integral is

(a)Trapezoidal rule states that

We have a=0, b=, n=30.

Therefore

Divide interval [0, into n=30 subintervals of length with the following endpoints

Now, we just evaluate function at those endpoints:

….

Finally, just sum up above values and multiply by

Answer :

Step 2</p>

We have a=0, b=, n=60.

Therefore

Divide interval [0, into n=60 subintervals of length with the following endpoints

Now, we just evaluate function at those endpoints:

….

Finally, just sum up above values and multiply by

Answer :