   Chapter 8.3, Problem 28E

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.28. y = sin x, y = 0, 0 ≤ x ≤ π

To determine

To sketch: The region bounded by the given curve, to visually estimates the centroid location and to find the exact coordinates of the centroid is (x¯,y¯).

Explanation

Given:

The equation of curve is y=sinx.

The region lies between 0 to π.

Calculation:

Plot a graph for the curve y=sinx and x=0 using the calculation as follows.

Calculate y value using equation (1)

Substitute 0 for x in equation (1).

y=sin0=0

Hence, the co-ordinate (x,y) is (0,0).

Substitute π2 for x in equation (1).

y=sin(π2)=1

Hence, the co-ordinate (x,y) is (π2,1).

Substitute π for x in Equation (1).

y=sin(π)=0

Hence, the co-ordinate (x,y) is (π,0).

By the use of the above points the region is drawn as shown below in Figure 1.

From the Figure 1 the coordinate (x¯) of the region is π2 and the coordinate (y¯) of the region is 0.4.

Hence, the location (x¯,y¯) of the region bounded by the given curve is (π2,0.4)_.

Calculate the area of the region:

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

Substitute 0 for a, π for b, and sinx for [f(x)] in equation (2).

A=0πsinxdx (2)

Integrate the above equation (2).

A=[cosx]0π=[cos(π)cos0]=(11)=2

Calculate the (x¯) coordinate of centroid:

x¯=1Aabx[f(x)]dx (3)

Substitute 0 for a, π for b, 2 for A, and sinx for [f(x)] in equation (3).

x¯=120πxsinxdx (4)

Integrate equation (4)

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