Chapter 8, Problem 45RE

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### 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. ∫ 2 3 2 1 + x 2 d x

To determine

To calculate: The approximate area of the integral 2321+x2dx using Trapezoidal and Simpson’s Rule.

Explanation

Given:

The provided integral is

âˆ«2321+x2dx

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 Î”x=bâˆ’a4. For the provided integral a=2 and

b=3.

So,

Î”x=3âˆ’24=14

And the provided function is f(x)=2x2+1

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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