#### To determine

**To find:**

The volume generated by rotating the region R1 about the line AB 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 is A=π radius2

ii. The volume of the solid revolution about the y-axis is

V= ∫abA(y)dy

**2) Given:**

**Figure:**

**3) Calculation:**

Here, OB appears to be the line y=x and AB appears to be the line x=1.

Rotating R1 about AB that is the line x=1, the cross section is perpendicular to the y-axis and it is a disc with the radius r=1-y because it is the distance from the line OB (y=x) to the axis of the rotation line AB (x=1).

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

By using the concept,

Substituting r=1-y, the volume is,

V= ∫01π1-y2dy

V=π∫011-2y+y2dy

By integrating,

V= πy-y2+y3310

V=π1-1+13-0

V=π3

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

V= π3

**Conclusion:**

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

V= π3