To determine

To find:

The volume generated by rotating the region R3 about line  OA.

Answer

V=π3

Explanation

1)  Concept:

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,

V= ∫abA(x)dx

2) Given:

Figure:

3) Calculation:

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,

V=π∫01x42-x2dx

V= π∫01x-x2dx

V=π23x32-x3310

V= π23-13-0=π3

Therefore,

The volume generated by rotating the region R3 about line OA is,

V= π3

Conclusion:

The volume generated by rotating the region R3 about line OA is,

V= π3