   Chapter 8.3, Problem 25E

Chapter
Section
Textbook Problem

Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.25. y = 2x, y = 0, x = 1

To determine

To sketch: The region bounded by the given curve and visually estimates the centroid location.

To find: The exact coordinates of the centroid (x¯,y¯).

Explanation

Given:

The equation of curve is y=2x.

The region lies between 0 to 1.

Calculation:

Show the equation as below:

y=2x (1)

Plot a graph for the equation y=2x and x=1 using the calculation as follows:

Calculate y value using Equation (1)

Substitute 0 for x in Equation (1).

y=2(0)=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=2(1)=2

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

Draw the region as in Figure 1.

Refer to Figure 1

The coordinate (x¯) of the region is 0.7.

The coordinate (y¯) of the region is 0.7.

Hence, the location (x¯,y¯) of the region bounded by the given curve is (0.7,0.7)_.

Calculate the area of the region:

A=ab[f(x)]dx (2)

Substitute 0 for a, 1 for b, and 2x for [f(x)] in Equation (2).

A=012xdx (3)

Integrate Equation (3).

A=[2x1+11+1]01=[2x22]01=[(1)2(0)2]=1

Calculate the (x¯) coordinate of centroid:

x¯=1Aabx[f(x)]dx (4)

Substitute 0 for a, 1 for b, 1 for A, and 2x for [f(x)] in Equation (4).

x¯=1101x(2x)dx=1012x2dx (5)

Integrate Equation (5)

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