#### To determine

**To find:**

The volume generated by rotating the region R3 about the line AB by using the figure.

#### Answer

V=1745π

#### 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:**

Rotating R3 about AB that is about line x=1, so the axis of rotation is x=1 .

Here, the cross section is perpendicular to the y-axis and it is a washer.

The outer radius is 1-y4, since the distance from x=y4 to the axis of rotation x=1 and

the inner radius is 1-y, since the distance from x=y to the axis of rotation x=1.

The solid lies between y=0 and y=1.

By using the concept,

V=π∫011-y42-1-y2dy

V= π∫01(1-2y4+y8-(1-2y+y2)) dy

V= π∫01-2y4+y8+2y-y2dy

V=π-2y55+y99+y2-y3310

V= π-25+19+1-13-0

V=17π45

Therefore, the volume generated by rotating the region R3 about the line AB is

V= 17π45

**Conclusion:**

The volume generated by rotating the region R3 about the line AB is

V= 17π45