   Chapter 8.3, Problem 77E

Chapter
Section
Textbook Problem

# Volume and Centriod In Exercises 77 and 78, for the region bounded by the graphs of the equations, find (a) the volume of the solid generated by revolving the region about the x -axis and (b) the centroid of the region. y = sin x ,   y = 0 ,   x = 0 ,   x = π

(a)

To determine

To calculate: Volume of the solid generated by revolving the region bounded by following graph of equation y=sinx,y=0,x=0,x=π about x-axis.

Explanation

Given:

The equations:y=sinx,y=0,x=0,x=π.

Formula used:

The volume of a solid by disk method

V=πab[R(x)]2dx.

Calculation:

Consider the following equations

y=sinx,y=0,x=0,x=π.

Nowuse the cylindrical shell methodto find the volume of a solid of revolution:

Plot the curves y=sinx,y=0 and shade the bounded region between them.

Now use disk method to calculate the volume of a solid,

V=πab[R(x)]2dx

Let’s assume that f(x)=sinx and g(x)=0

We can find the radius by subtracting g(x) from f(x)

(b)

To determine

To calculate: The centroid of the region bounded by the following graphs of equations y=sinx,y=0,x=0,x=π.

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