To determine

To find:

The volume generated by rotating the region R1 about the line  BC by using the figure.

Answer

V=23π

Explanation

1) Concept:

i. If the cross section is a washer with the inner radius rin and the outer radius rout, then the area of the washer is obtained by subtracting the area of the inner disk from the area of the outer disk.

A=π outer radius2 -π inner radius2

ii. The volume of the solid revolution about the x-axis is

V= ∫abA(x)dx

2) Given:

Figure:

3) Calculation:

Here, OB appears to be the line y=x and BC appears to be the line y=1.

Rotating R1 about BC, that is, the line  y=1, the cross section is perpendicular to the x-axis and it is a washer.

The outer radius is  1-0=1, since it is the distance from the x axis (y=0) to the axis of rotation  y=1.

And the inner radius is  1-x, since it is the distance from the y=x  to the axis of rotation y=1

The solid lies between x=0 and  x=1.

By using the concept,

V=π∫0112-1-x2dx

V= π∫011-1-2x+x2dx=π∫012x-x2dx

After integrating,

V=πx2-x3310

V= π1-13=23π

The volume generated by rotating the region R1 about the line BC is

V=23π

Conclusion:

The volume generated by rotating the region R1 about the line BC is

V=23π