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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
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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