Question

Use the error formulas to estimate the errors in approximating the integral, with n = 4, using (a) the Trapezoidal Rule and (b) Simpson's Rule.

Answer 1

Step 1:

(a)

The integral function is .

Let

Error in Trapezoidal rule: 

If has a continuous second derivative on , then the error approximating the integral by Trapezoidal Rule is   where .

First derivative is

Second derivative is

The function has a continuous second derivative on the interval .

The maximum value of on the interval is .

Approximate error in trapezoidal rule:

Solution:

(a) Error in Trapezoidal rule .

Answer 2

Step 2:

(b)

The integral function is .

Error in Simpsons Rule: 

If has a continuous fourth derivative on , then the error approximating the integral by Simpsons Rule is   where .

First derivative is .

Second derivative is .

Third derivative is .

Fourth derivative is .

The function has a continuous fourth derivative on the interval .

The maximum value of on the interval is .

Approximate error in Simpsons rule:

Solution:

(b) Error in Simpsons rule .