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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

If x ¯ is the x-coordinate of the centroid of the region that lies under the graph of a continuous function f, where axb, show that

a b ( c x + d ) f ( x ) d x = ( c x ¯ + d ) a b f ( x ) d x

To determine

To prove: The integral function as ab(cx+d)f(x)dx=(cx¯+d)abf(x)dx

Explanation

Given:

Show the integral function as follows:

ab(cx+d)f(x)dx=(cx¯+d)abf(x)dx

The integral function of left hand side (LHS) is ab(cx+d)f(x)dx.

Calculation:

The (x¯) coordinate of centroid is expressed as follows:

x¯=1Aabxf(x)dx (1)

Rearrange Equation (1).

x¯A=abxf(x)dx

The area of the function as follows:

A=abf(x)dx

The integral function as follows:

ab(cx+d)f(x)dx=ab[cxf(x)+df(x)]dx=abcxf(x)dx+abdf(x)dx=cabxf(x)dx+dabf(x)dx (2)

Substitute x¯A for abxf(x)dx in Equation (2)

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