   Chapter 8.3, Problem 43E

Chapter
Section
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:

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)

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

In Exercises 2340, find the indicated limit. 39. limx1x2+82x+4

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Find f. f(t) = 4/(1 + t2), f(1) = 0

Single Variable Calculus: Early Transcendentals, Volume I

State Keplers Laws.

Multivariable Calculus

In Problems 5-10, find the slope of the line passing through the given pair of points.

Mathematical Applications for the Management, Life, and Social Sciences

ex(ex)2 = a) ex3 b) e3x c) 3ex d) e4x

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

Study Guide for Stewart's Multivariable Calculus, 8th

Explain how the mean and the standard deviation describe a distribution of scores.

Research Methods for the Behavioral Sciences (MindTap Course List) 