   Chapter 8, Problem 15RE

Chapter
Section
Textbook Problem

Find the centroid of the region bounded by the given curves.15. y = 1 2 x ,   y = x

To determine

To find: The centroid of the region bounded by the curves.

Explanation

Given:

The equations of the curves are y=12x and y=x .

Calculation:

Procedure to sketch the region bounded by the two curves is explained below:

• Draw the graph for the function y=12x by substituting different values for x.
• Similarly in the same graph plot for the function y=x by substituting different values for x.
• Shade the region lies between the intersecting points of the curves.

The region enclosed by the curves y=12x and y=x is shown in Figure 1.

Refer Figure 1.

The curve intersects at x=0 and x=4 .

The expression to find the area of the shaded region is shown below:

A=ab[f(x)g(x)]dx (1)

Here, the lower limit is a, the upper limit is b, the top curve function is f(x) , and the bottom curve function is g(x) .

Substitute 0 for a, 4 for b, x for f(x) , and 12x for g(x) in Equation (1).

A=04(x12x)dx=[x323212x22]04=(23(4)3214(4)2)(0)=1634

=16123=43

Calculate the x-coordinate of the centroid (x¯) using the relation:

x¯=1Aabx[f(x)g(x)]dx (2)

Substitute 43 for A, 0 for a, 4 for b, x for f(x) , and 12x for g(x) in Equation (2)

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