Chapter 8.3, Problem 43E

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

Chapter
Section

### 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 a ≤ x ≤ b, 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:

âˆ«ab(cx+d)f(x)dx=(cxÂ¯+d)âˆ«abf(x)dx

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

Calculation:

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

xÂ¯=1Aâˆ«abxf(x)dx (1)

Rearrange Equation (1).

xÂ¯A=âˆ«abxf(x)dx

The area of the function as follows:

A=âˆ«abf(x)dx

The integral function as follows:

âˆ«ab(cx+d)f(x)dx=âˆ«ab[cxf(x)+df(x)]dx=âˆ«abcxf(x)dx+âˆ«abdf(x)dx=câˆ«abxf(x)dx+dâˆ«abf(x)dx (2)

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

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