#### 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=3π4

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=1x, y=0, x=1, x=4; About the x- axis

**3) Calculation:**

The solid obtained by rotating the region bounded by

y=1x, y=0, x=1, x=4 about the x- axis is as 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 a solid is disk with radius 1x ,

So its cross sectional area becomes,

Ax=π1x2=πx-2

The solid lies between x=1 and x= 4

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

V= ∫14Axdx=∫14πx-2dx

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

=π-x-114=π-1x14

By substituting limits of integration,

=π -14--1

=π-14+1

=π -1+44

V=3π4

Therefore,

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

V=3π4

**Conclusion:**

Therefore,

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

V=3π4

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