   Chapter 8.3, Problem 37E

Chapter
Section
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=x3x and y=x21.

Calculation:

Show the equations as below:

y=x3x (1)

y=x21 (2)

Plot a graph for the equations y=x3x and y=x21 using the calculation as follows:

Calculate y value using Equation (1)

Substitute 0 for x in Equation (1).

y=(0)30y=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)31y=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)21=1

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

Calculate x value using Equation (2).

Substitute 1 for y in Equation (2).

y=(1)21=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 x¯ is 15

The coordinate y¯ is 1235

Hence, the centroid of the region is (15,1235)_.

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

f(x)=x3x

g(x)=x21

Calculate the area of the region:

A=ab[f(x)g(x)]dx (3)

Substitute (1) for a, 1 for b, (x3x) for [f(x)], and (x21) for [g(x)] in Equation (3).

A=11[(x3x)(x21)]dx=11(x3x2x+1)dx (4)

Integrate Equation (4).

A=[x3+13+1x2+12+1x1+11+1+x]11=[x44x33x22+x]11=[(1)44(1)33(1)22+(1)][(1)44(1)33(1)22+(1)]

=512(1112)=43

Calculate the (x¯) coordinate of centroid:

x¯=1Aabx[f(x)g(x)]dx (5)

Substitute (1) for a, 1 for b, 43 for A, (x3x) for [f(x)], and (x21) for [g(x)] in Equation (5).

x¯=14311x[(x3x)(x21)]dx=3411x(x3x2x+1)dx=3411(x4x3x2+x)dx (6)

Integrate Equation (6)

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