#### To determine

**To find:**

The volume generated by rotating the region R1 about the line OC by using the figure.

#### Answer

V=2π3

#### Explanation

**1) Concept:**

i. If the cross section is a washer with the inner radius rin and the outer radius rout, then the area of the 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 the y-axis is

V= ∫abA(y)dy

**2) Given:**

**Figure:**

**3) Calculation:**

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

Rotating R1 about OC that is the line x=0, the cross section is perpendicular to the y -axis and it is a washer with the inner radius y and the outer radius 1.

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

Rotation of R1 is about OC, that is, about line x=0.

By using concept,

V= ∫01π12-y2dy

By integrating,

V=πy-y3310

V=π1-13=2π3

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

V=2π3

**Conclusion:**

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

V=2π3