   Chapter 4.2, Problem 64E

Chapter
Section
Textbook Problem

# Use Property 8 of integrals to estimate the value of the integral. ∫ π 2 π ( x − 2 sin x )   d x

To determine

To estimate:

The value of thegiven integral

Explanation

1) Concept:

Use comparison property (8)  of the integral.

If mfxM for axb, then mb-aabfxdxMb-a

2) Given:

π2π(x-2sinx)dx

3) Calculation:

Here,fx=x-2sinx,  a=π,   b=2π

Since this function is neither increasing nor decreasing over the given interval, we have to find its absolute maximum and absolute minimum over the given interval using thederivative test.

So, differentiate with respect to x

f'x=1-2cosx

f'x=0 at only x=5π3 in given interval

Therefore, to find the minimum and maximum value of fx  compute fx at x=5π3, π, 2π

f5π3=5π3-2sin5π3

f5π3=5π3-2-32

f5π3=5π3+3

f5π36.97

fπ=π-2sinπ

fπ=π-20

fπ=π

fπ3

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