   Chapter 4.2, Problem 11E

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 2 x x + 1 d x ,    n = 5

To determine

To approximate:

The integral 02xx + 1dx with n=5, by using the midpoint rule and then round the answer to four decimal places.

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:

02xx + 1dx,  n=5

4) Calculation:

Here,n=5, a=0, and b=2.

So, the width of subinterval is

x=b-an

Substitute values of a, b, and n in above step.

x=2-05

x=25

Use the width x=25 to form the subintervals.

0,25, 25,45, 45,65, 65,85 and 85, 2

Now, find the midpoints of all subintervals. It becomes

x1-=12x0+x1=120+25=1225=15

x2-=12x1+x2=1225+45=1265=35

x3-=12x2+x3=1245+65=12

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