#### 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= 943π

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 area A=π radius2

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

iii. V= ∫abA(x)dx

**2) Given:**

y=25-x2, y=0, x=2, x=4 about the x- axis.

**3) Calculation:**

The region is bounded by y=25-x2, y=0, x=2, x=4 and rotated about the x- axis is shown in figure:

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 25-x2.

So, its cross sectional area becomes

Ax=π25- x22=π25-x2

The solid lies between x=2 and x= 4

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

V= ∫24Axdx=∫24π25-x2dx

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

V=π25x-x3324

By substituting the limits of integration,

=π 25(4)-433-25(2)-233

=π100-643-50+83

=π 50-563=π150-563

V=94π3

**Conclusion:**

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

V=94π3

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