#### To determine

**To find:**

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

#### Answer

V=26π3

Sketch of bounded region, the solid, and a typical disk:

#### 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 area A=π radius2

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

V= ∫abA(x)dx

**2) Given:**

y=x+1, y=0, x=0, x=2, rotation about the x- axis

**3) Calculation:**

The region is bounded by y=x+1, y=0, x=0, x=2 and rotated about the x- axis is shown below:

Here, the region is rotated about the x – axis, so the cross-section is perpendicular to the x-axis.

A cross section of the solid is a disk with radius x+1.

So, its cross sectional area becomes

Ax=πx+12=πx2+2x+1

The solid lies between x=0 and x= 2.

So, the volume of the solid of revolution about the x-axis is

V= ∫02Axdx=∫02πx2+2x+1dx

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

V=π13x3+x2+x02

By substituting the limits of integration,

=π 13·23+22+2-13·03+02+0

=π83+4+2

=π 8+12+63

V=26π3

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

V=26π3

**Conclusion:**

The volume of the solid obtained by rotating the region bounded by the given curves about the x axis is

V=26π3

Sketch of bounded region, the solid and a typical disk: