To determine

To find:

The volume generated by rotating the region R1 about theline  OA by using the figure.

Answer

V= π3

Explanation

1)  Concept:

i. If the cross section is a disc and the radius of the disc is in terms of x  or y, then the area 

A=π radius2

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

V= ∫abA(x)dx

2) Given:

Figure:

3) Calculation:

Here, the line OB appears to be the line y=x and line OA appears to be the line  y=0.

On rotating R1 about OA that is the line  y=0, the cross section is perpendicular to the x-axis. And it is a disc with radius r=x because it is the distance from the line OB(y=x)  to the axis of rotationOA (y=0).

From the figure, the solid lies between x=0 and  x=1.

By using the concept,

V=∫01πr2 dx

Substituting r=x,

V=π∫01x2 dx

By integrating,

V=πx3310

After applying the limits,

V=π13-0=π3

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

V=π3

Conclusion:

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

V=π3