#### To determine

**To find:**

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

#### Answer

V=415π

#### 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 x-axis is

V= ∫abA(x)dx

**2) Given:**

**Figure:**

**3) Calculation:**

Rotating R3 about BC, that is, about line y=1, so the cross section is perpendicular to the x-axis and it is a washer.

Here, the outer radius is 1-x, since it is the distance from y=x to the axis of rotation

y=1 and the inner radius 1-x4, since it is the distance from y=x4 to the axis of rotation y=1.

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

By using the concept,

V=π∫01(1-x)2-1-x42dx

V= π∫011-2x+x2-1-2x4+xdx=π∫01-2x+x2+2x14-x12dx

After integrating,

V=π-x2+x33+2·45x54-23x3210

V= π-1+13+85-23-0

V=415π

Therefore,

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

V=415π

**Conclusion:**

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

V=415π