Chapter 8, Problem 16RE

### 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.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 Ï€4 to 3Ï€4 .

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 Ï€4 to 3Ï€4 .
• Shade the region lies between the region Ï€4 and 3Ï€4 .

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 (xÂ¯) is Ï€2 .

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

A=âˆ«ab[f(x)]â€‰dx (1)

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

Substitute Ï€4 for a, 3Ï€4 for b, and sinx for f(x) in Equation (1).

A=âˆ«Ï€43Ï€4sinxâ€‰dx=[âˆ’cosx]Ï€43Ï€4=âˆ’cos(3Ï€4)+cos(3Ï€4)=12+12

=22=2

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

yÂ¯=1Aâˆ«ab12{[f(x)]2}â€‰dx (2)

Substitute 2 for A, Ï€4 for a, 3Ï€4 for b, and sinx for f(x) in Equation (2)

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