   Chapter 4.2, Problem 10E

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. ∫ 0 1 x 3 + 1   d x ,    n = 5

To determine

To approximate:

The integral 01x3+1 dx with n=5 by using the midpoint rule then round the answer to four decimal places

Explanation

1) Concept:

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

2) Themidpoint 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:

01x3+1 dx,  n=5

4) Calculation:

Here,n=5, a=0 and b=1.

So, the width of subinterval is

x=b-an

x=1-05

x=15

Use width x=15 to form the subintervals.

0, 15, 15,25, 25,35, 35,45 and 45, 1

Now, find the midpoints of all subintervals. It becomes

x1-=12x0+x1=120+15=1215=110

x2-=12x1+x2=1215+25=1235=310

x3-=12x2+x3=122

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