#### 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