Chapter 8.3, Problem 28E

### 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

# 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 Ï€.

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 Ï€2 for x in equation (1).

y=sin(Ï€2)=1

Hence, the co-ordinate (x,y) is (Ï€2,1).

Substitute Ï€ for x in Equation (1).

y=sin(Ï€)=0

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

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

From the Figure 1 the coordinate (xÂ¯) of the region is Ï€2 and the coordinate (yÂ¯) of the region is 0.4.

Hence, the location (xÂ¯,yÂ¯) of the region bounded by the given curve is (Ï€2,0.4)_.

Calculate the area of the region:

A=âˆ«ab[f(x)]â€‰dx (1)

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

A=âˆ«0Ï€sinxâ€‰dx (2)

Integrate the above equation (2).

A=[âˆ’cosx]0Ï€=âˆ’[cos(Ï€)âˆ’cos0]=âˆ’(âˆ’1âˆ’1)=2

Calculate the (xÂ¯) coordinate of centroid:

xÂ¯=1Aâˆ«abx[f(x)]â€‰dx (3)

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

xÂ¯=12âˆ«0Ï€xsinxâ€‰dx (4)

Integrate equation (4)

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