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Chapter 8, Problem 47RE
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### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

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### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Using the Trapezoidal Rule and Simpson's Rule In Exercises 45–48, approximate the definite integral using the Trapezoidal Rule and Simpson’s Rule with n = 4. Compare these results with the approximation of the integral using a graphing utility. ∫ 0 π / 2 x cos x   d x

To determine

To calculate: The approximate area of the integral 0π2xcosxdx using Trapezoidal and Simpson’s Rule.

Explanation

Given:

The provided integral is

âˆ«0Ï€2xcosxdx

And number of subintervals to be used n=4

Formula used:

If f(x) is continuous on [a,b] then Trapezoidal rule for approximating âˆ«abf(x)dx is

âˆ«abf(x)dxâ‰ˆbâˆ’a2n[f(x0)+2f(x1)+2f(x2)+â€¦+2f(xnâˆ’1)+f(xn)]

Where n is be an even integer and is called number of subintervals.

And

Simpsonâ€™s Rule for approximating âˆ«abf(x)dx is

âˆ«abf(x)dxâ‰ˆbâˆ’a3n[f(x0)+4f(x1)+2f(x2)+4f(x3)â€¦+2f(xnâˆ’2)+4f(xnâˆ’1)+f(xn)]

Calculation:

When n=4, then the width of each subinterval is Î”x. For the provided integral a=0 and

b=Ï€2.

So,

Î”x=Ï€2âˆ’04=Ï€8

And the provided function is f(x)=xcosx

Now apply Trapezoidal rule. That is,

âˆ«abf(x)dxâ‰ˆbâˆ’a2n[f(x0)+2f(x1)+2f(x2)+â€¦+2f(xnâˆ’1)+f(xn)]

Put values of f(x),a and b

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