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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Use the properties of integrals to verify the inequality without evaluating the integrals.

π 12 π / 6 π / 3 sin x d x 3 π 12

To determine

To verify: The inequality of the integral function π/6π/3sinxdx using the properties of integrals.

Explanation

Given information:

The integral function is π/6π/3sinxdx.

The region lies between x=π6 and x=π3.

Consider the function f(x)=sinx (1)

Apply property 8 of integrals:

If mf(x)M for axb, then m(ba)abf(x)dxM(ba).

To find the absolute maximum value M as shown below:

Calculate absolute maximum value M using Equation (1).

M=f(x) (2)

Substitute π3 for x in Equation (2).

M=f(π3)=sin(π3)=32

To find the absolute minimum value m as shown below:

Calculate absolute minimum value using Equation (1).

m=f(x) (3)

Substitute π6 for x in Equation (3).

m=f(π6)=sin(π6)=12

The value of x is increasing at the interval (π6,π3)

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