menu
Hit Return to see all results
Chapter 8.3, Problem 37E
BuyFindarrow_forward

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

8th Edition
James Stewart
ISBN: 9781305270336

#### Solutions

Chapter
Section
BuyFindarrow_forward

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

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

# Find the centroid of the region bounded by the curves y = x3 − x and y = x2 − 1. Sketch the region and plot the centroid to see if your answer is reasonable.

To determine

To find: The centroid of the region bounded by the curves and sketch the region and plot the centroid.

Explanation

Given:

The equations are y=x3âˆ’x and y=x2âˆ’1.

Calculation:

Show the equations as below:

y=x3âˆ’x (1)

y=x2âˆ’1 (2)

Plot a graph for the equations y=x3âˆ’x and y=x2âˆ’1 using the calculation as follows:

Calculate y value using Equation (1)

Substitute 0 for x in Equation (1).

y=(0)3âˆ’0y=0

Hence, the co-ordinate of (x,y) is (0,0).

Calculate y value using Equation (1)

Substitute 1 for x in Equation (1).

y=(1)3âˆ’1y=0

Hence, the co-ordinate of (x,y) is (1,0).

Calculate x value using Equation (2)

Substitute 0 for y in Equation (2).

y=(0)2âˆ’1=âˆ’1

The co-ordinate of (x,y) is (0,âˆ’1).

Calculate x value using Equation (2).

Substitute 1 for y in Equation (2).

y=(1)2âˆ’1=0

The co-ordinate of (x,y) is (1,0).

Similarly calculate the coordinate values up to bound the region in the graph.

Draw the region as shown in Figure 1.

Refer to Figure 1.

The coordinate xÂ¯ is âˆ’15

The coordinate yÂ¯ is âˆ’1235

Hence, the centroid of the region is (âˆ’15,â€‰âˆ’1235)_.

Consider the f(x) and g(x) as follows:

f(x)=x3âˆ’x

g(x)=x2âˆ’1

Calculate the area of the region:

A=âˆ«ab[f(x)âˆ’g(x)]dx (3)

Substitute (âˆ’1) for a, 1 for b, (x3âˆ’x) for [f(x)], and (x2âˆ’1) for [g(x)] in Equation (3).

A=âˆ«âˆ’11[(x3âˆ’x)âˆ’(x2âˆ’1)]dx=âˆ«âˆ’11(x3âˆ’x2âˆ’x+1)dx (4)

Integrate Equation (4).

A=[x3+13+1âˆ’x2+12+1âˆ’x1+11+1+x]âˆ’11=[x44âˆ’x33âˆ’x22+x]âˆ’11=[(1)44âˆ’(1)33âˆ’(1)22+(1)]âˆ’[(âˆ’1)44âˆ’(âˆ’1)33âˆ’(âˆ’1)22+(âˆ’1)]

=512âˆ’(âˆ’1112)=43

Calculate the (xÂ¯) coordinate of centroid:

xÂ¯=1Aâˆ«abx[f(x)âˆ’g(x)]â€‰dx (5)

Substitute (âˆ’1) for a, 1 for b, 43 for A, (x3âˆ’x) for [f(x)], and (x2âˆ’1) for [g(x)] in Equation (5).

xÂ¯=143âˆ«âˆ’11x[(x3âˆ’x)âˆ’(x2âˆ’1)]â€‰dx=34âˆ«âˆ’11x(x3âˆ’x2âˆ’x+1)â€‰dx=34âˆ«âˆ’11(x4âˆ’x3âˆ’x2+x)â€‰dx (6)

Integrate Equation (6)

### 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

## Additional Math Solutions

#### Find more solutions based on key concepts

Show solutions add

#### let f(x) = x3 + 5, g(x) = x2 2, and h(x)= 2x + 4, Find the rule for each function. 1. f + g

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

#### In Problems 27 â€“ 30, solve for y in terms of x. 28.

Mathematical Applications for the Management, Life, and Social Sciences

#### True or False: f(x) = tan x is differentiable at x=2.

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

#### Draw a scatterplot for which r = 1.

Introduction To Statistics And Data Analysis