   Chapter 8, Problem 16RE

Chapter
Section
Textbook Problem

Find the centroid of the region bounded by the given curves.16. y = sin x, y = 0, x = π/4, x = 3π/4

To determine

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

Explanation

Given:

The equations of the curves are y=sinx and y=0 .

The region lies between π4 to 3π4 .

Calculation:

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

• Draw the graph for the function y=sinx by substituting values from π4 to 3π4 .
• Shade the region lies between the region π4 and 3π4 .

The region enclosed by the curve y=sinx is shown in Figure 1.

Refer Figure 1.

The curve is symmetric about y-axis, the x-coordinate of the centroid (x¯) is π2 .

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

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

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

Substitute π4 for a, 3π4 for b, and sinx for f(x) in Equation (1).

A=π43π4sinxdx=[cosx]π43π4=cos(3π4)+cos(3π4)=12+12

=22=2

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

y¯=1Aab12{[f(x)]2}dx (2)

Substitute 2 for A, π4 for a, 3π4 for b, and sinx for f(x) in Equation (2)

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