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