#### To determine

**To find:**

The volume of solid obtained by rotating the region bounded by the given curves about the line x=2 and sketch the region, the solid, and a typical disk or washer.

#### Answer

V=3π5

The sketch of the bounded region, the solid, and a typical washer:

#### 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:**

y=x3, y=0,x=1; about the line x=2

**3) Calculation:**

The region bounded by y=x3, y=0,x=1 and the solid obtained by rotation about the line x=2 are shown below.

Here, the region is rotated about the line x=2, so the cross-section is perpendicular to y-axis.

A cross section of the solid is the washer with the outer radius 2-y3 and the inner radius is 2-1=1.

So, its cross sectional area is

Ay=πouter radius2-πinner radius2 =π2-y32-π12

=π4-4y13+y23-1

A(y)=π3-4y13+y23

The region of integration is bounded by y=x3, y=0 and x=1

So one boundary point is given by,

y=13

y=1

The solid lies between y=0 and y= 1

Therefore, the volume of the solid revolution about the line x=2 is

V= ∫01Aydy=∫01π3-4y13+y23dy

By using the fundamental theorem of calculus part 2 and the power rule of integration,

V=π3y-4y4343+y535301

=π3y-3y43+3y53501

By substituting the limits of integration,

=π 3(1)-3(1)43+3(1)535-3(0)-3(0)43+3(0)535

=π 3-3+35-0

V=3π5

Therefore, the volume of the solid obtained by rotating the region bounded by the given curves about the line x=2 is

V=3π5

**Conclusion:**

Therefore, the volume of the solid obtained by rotating the region bounded by the given curves about the line x=2 is,

V=3π5

The sketch of the bounded region, the solid, and a typical washer: