The volume generated by rotating the region R3 about line OA.
i. If the cross section is a washer with inner radius rin and outer radius rout, then area of 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 x-axis is,
Here, OB appears to be the line y=x and OA appears to be the line y=0.
Rotating R3 about OA that is the line y=0, so the cross section is perpendicular to the x-axis and it is a washer.
Thus, the outer radius is the distance from the curve y= x4 to the axis of rotation OA
(y=0) and the inner radius is the distance from the line OB (y=x) to the axis of rotation OA (y=0). So the outer radius = x4 and inner radius is x.
The solid lies between x=0 and x=1.
By using concept,
The volume generated by rotating the region R3 about line OA is,