   Chapter 4.2, Problem 12E

Chapter
Section
Textbook Problem

# Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. ∫ 2 π x sin 2 x   d x ,    n = 4

To determine

To approximate:

The integral 0πx sin2xdx with n=4 by using the midpoint rule.

Solution:

Approximately 2.4674

Explanation

1) Concept:

Use the midpoint rule to approximate the value of the integral.

2) Midpoint Rule:

abf(x)dxi=1nfxi-x=x[fxi-+ .+ fxn-]

where x=b - an, n is the number of the subintervals and  xi-=12xi-1+xi is the midpoint of [xi-1, xi]

3) Given:

0πx sin2xdx,  n=4

4) Calculation:

Here,n=4, a=0, and b=π.

So, the width of the subinterval is

x=b-an

x=π-04

x=π4

Use the width x=π4 to form the subintervals

0,π4, π4, 2π4, 2π4, 3π4 and 3π4, π

Now, find the midpoints of all subintervals. Thus we have

x1-=12x0+x1=120+π4=12π4=π8

x2-=12x1+x2=12π4+2π4=123π4=3π8

x3-=12x2+x3

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