Chapter 8.3, Problem 29E

### 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

# 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=âˆ«ab[f(x)âˆ’g(x)]â€‰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=âˆ«01(xâˆ’x2)â€‰dx=âˆ«01(x12âˆ’x2)â€‰dx (4)

Integrate Equation (4).

A=[x12+112+1âˆ’x2+12+1]01=[23x32âˆ’13x3]01=(23(1)32âˆ’13(1)3)âˆ’0=13

Calculate the (xÂ¯) coordinate of centroid:

xÂ¯=1Aâˆ«abx[f(x)âˆ’g(x)]â€‰dx (5)

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

xÂ¯=113âˆ«01x(xâˆ’x2)â€‰dx=3âˆ«01x(x12âˆ’x2)â€‰dx=3âˆ«01(x12+1âˆ’x2+1)â€‰dx=3âˆ«01(x32âˆ’x3)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