#### To determine

**To find:**

The volume generated by rotating the region R2 about the line AB by using the figure.

#### Answer

V=13π45

#### 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 AB appears to be the line x=1 and the line OC appears to be the line x=0.

Rotating R2 about AB, that is, the line x=1, so the cross section is perpendicular to the y -axis and it is a washer.

Thus, the outer radius is the distance from the line OC (x=0) to the axis of rotation AB

(x=1) and the inner radius is the distance from the curve y= x4 to the axis of rotation AB (x=1). So, the outer radius is 1-0=1 and inner radius is 1-y4.

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

By using the concept,

V=π∫0112-1-y42dy

V= π∫011-(1-2y4+y8) dy

V= π∫012y4-y8dy

After integrating,

V=π2y55-y9910

V= π25-19-0=13π45

Therefore, the volume generated by rotating the region R2 about line AB is

V= 13π45

**Conclusion:**

The volume generated by rotating the region R2 about the line AB is

V= 13π45