# Approximate Integrals using the Trapezoidal and Simpson's Rule

1. Approximate the integrals using the Trapezoidal rule.

a) Integral from -0.5 to 0 x ln(x+1) dx

b) Integral from 0.75 to 1.3 ((sin x)2 - 2x sin x +1) dx

2. Find a bound for error in question 1. using the error formula, and compare this to the actual error.

3. Repeat question 1. using Simpson's rule

4. Repeat question 2. using Simpson's rule and the result of question 3

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#### Solution Preview

1. Approximate the integrals using the Trapezoidal rule.

a) Integral from -0.5 to 0 x ln(x+1) dx

b) Integral from 0.75 to 1.3 ((sin x)2 - 2x sin x +1) dx

2. Find a bound for error in question 1. using the error formula, and compare this to the actual error.

3. Repeat question 1. Using Simpson's rule

4. Repeat question 2. Using Simpson's rule and the result of question 3

Solution:

a)

Trapezoidal formula: I: = , where h denotes the step size.

h = b-a/n, suppose if we assume n = 5 sub intervals then we have the step size computed as follows:

a = -0.5, b = 0, then h = 0 - (-0.5)/5 = 0.5/5 = 0.1

Hence the step size h = 0.1

The integrand f(x) = xln(x+1)

When x = -0.5, f(-0.5) = -0.5ln(1-0.5) = -0.5ln(0.5) = 0.34657

When x = 0, f(0) = 0

Hence the integral due to Trapezoidal rule:

I: =

b)

Trapezoidal formula: I = , where h denotes the step size.

h = b-a/n, again for n = 5, a=0.75, b = 1.3, h = (1.3-0.75)/5 = 0.55/5 = 0.11

Hence the step size ...

#### Solution Summary

Integrals are approximated using the Trapezoidal rule and Simpson's rule. The solution is detailed and well presented.