   Chapter 8, Problem 47RE

Chapter
Section
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

0π2xcosxdx

And number of subintervals to be used n=4

Formula used:

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

abf(x)dxba2n[f(x0)+2f(x1)+2f(x2)++2f(xn1)+f(xn)]

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

And

Simpson’s Rule for approximating abf(x)dx is

abf(x)dxba3n[f(x0)+4f(x1)+2f(x2)+4f(x3)+2f(xn2)+4f(xn1)+f(xn)]

Calculation:

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

b=π2.

So,

Δx=π204=π8

And the provided function is f(x)=xcosx

Now apply Trapezoidal rule. That is,

abf(x)dxba2n[f(x0)+2f(x1)+2f(x2)++2f(xn1)+f(xn)]

Put values of f(x),a and b

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