   Chapter 8.3, Problem 29E

Chapter
Section
Textbook Problem

Find the centroid of the region bounded by the given curves.29. y = x2, x = y2

To determine

To find: The centroid (x¯,y¯) of the region bounded by the given curve.

Explanation

Given:

The equations of curve are y=x2 and x=y2.

Calculation:

Show the equations as below:

y=x2 (1)

x=y2 (2)

Plot a graph for the equations y=x2 and x=y2 using the calculation as follows:

Calculate y value using Equation (1)

Substitute 0 for x in Equation (1).

y=02=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=12=1

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

Calculate x value using Equation (2)

Substitute 0 for y in Equation (2).

x=02=0

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

Calculate x value using Equation (2).

Substitute 1 for y in Equation (2).

x=12=1

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

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

Draw the region as in Figure 1.

Calculate the area of the region:

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

Rearrange Equation (2).

x=y2y=x

Substitute 0 for a, 1 for b, x for [f(x)], and x2 for [g(x)] in Equation (3).

A=01(xx2)dx=01(x12x2)dx (4)

Integrate Equation (4).

A=[x12+112+1x2+12+1]01=[23x3213x3]01=(23(1)3213(1)3)0=13

Calculate the (x¯) coordinate of centroid:

x¯=1Aabx[f(x)g(x)]dx (5)

Substitute 0 for a, 1 for b,13 for A, x for [f(x)], and x2 for [g(x)] in Equation (5).

x¯=11301x(xx2)dx=301x(x12x2)dx=301(x12+1x2+1)dx=301(x32x3)dx (6)

Integrate Equation (6)

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