   Chapter 4.R, Problem 45E

Chapter
Section
Textbook Problem

# Use the Midpoint Rule with n = 6 to approximate ∫ 0 3 sin ( x 3 ) d x

To determine

To approximate:

The given integral 03sin(x3)dx by using the midpoint rule with n=6

Explanation

1) Concept:

i) The midpoint rule: abfxdxi=1nf(xi-)x=x[fx1-+·····+fxn-]

where x=b-an  and  xi-=12xi-1+xi= midpoint of [xi-1,xi]

2) Given:

03sin(x3)dx

3) Calculation:

Given that the integral is 03sin(x3)dx. Applying the midpoint rule on it 03sin(x3)dxi=16f(xi-)x Here

x=b-an=3-06=36=12

Now divide the interval [0,3] into 6 subintervals with endpoints 0 , 12, 1, 32, 2,52, 3

Therefore, the six subintervals are

0,12, 12, 1, 1,32, 32, 2 , 2,52, 52, 3

And the midpoints xi- of these intervals are

Midpoint of 0,12=120+12=14

Midpoint of 12, 1=1212+1=34

Midpoint of 1,32=121+32=

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