#### To determine

**To find:**

The volume generated by rotating the region bounded by the given curves about x=1 using the cylindrical shells.

#### Answer

V=8π3unit3

#### Explanation

**1) Concept:**

i. If x is the radius of the typical shell, then the circumference =2πx and the height is y=f(x)

ii. By the shell method, the volume of the solid by rotating the region under the curve y=f(x) about x-axis from a to b is

V= ∫ab2πx f(x)dx

where 0≤a<b

**2) Given:**

y=4x-x2, y=3, rotate about x=1

**3) Calculation:**

As the region is bounded by y=4x-x2, y=3, and rotated about x=1, draw the region using the given curves.

The graph shows the region and the vertical cross section of cylindrical shell formed by the rotation about the line x=1.

It has the radius =x-1

Therefore, the circumference is 2π(x-1) and height is (4x-x2)-3

To find the point of intersection, equate both curves

4x-x2=3

x2-4x+3=0

This is a quadratic equation.

Solve by using the quadratic formula.

x=4±16-122=1, 3

So, a=1 and b=3

So, the volume of the given solid is

V= ∫ab2πx+1fxdx

=∫132πx-14x-x2-3dx

=2π∫13-x3+5x2-7x+3dx

=2π-x44+5x33-7x22+3x13

=2π-344+5333-7322+33--144+5133-7122+31

=2π43

=8π3

**Conclusion:**

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

V=8π3unit3