#### 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= 8π

The sketch of the 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 the area A=π radius2

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

V= ∫abA(x)dx

**2) Given:**

The region is bounded by y =x-1, y=0, x=5; about the x- axis.

**3) Calculation:**

The region is bounded by y=x-1, y=0, x=5 and solid is obtained by rotating it about the x- axis. It is shown below:

Here, the region 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=πx-1

The solid lies between x=1 and x= 5

Therefore, the volume of the solid is

V= ∫15Axdx=∫15πx-1dx

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

V=π12x2-x15

By substituting the limits of integration,

=π 1252-5-1212-1

=π252-5-12+1

=π 242-4=π(12-4)

V=8π

Therefore, the volume of the solid obtained by rotating the region bounded by the given curves about the x axis is V=8π

**Conclusion:**

The volume of the solid obtained by rotating the region bounded by the given curves about the x axis is V=8π.

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