#### To determine

**To find:**

The volume generated by rotating the region bounded by the given curves about y-axis using the cylindrical shells.

#### Answer

V=27π2unit3

#### Explanation

**1) Concept:**

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

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

V= ∫ab2πxf(x)dx

where 0≤a≤b

**2) Given:**

The region bounded by y=4x-x2, y=x rotated about the y- axis.

**3) Calculation:**

As the region is bounded by y=4x-x2, y=x rotated about the y- axis,

using the shell method, the typical approximating shell with the radius x is

Therefore, the circumference is 2πx and the height is 4x-x2-x.

that is, -x2+3x

To find the values of a and b, consider 4x-x2=x

4x-x2-x=0

-x2+3x=0

that is, x3-x=0

x=0 or x=3

Therefore, a=0 and b=3

So, the total volume is

V= ∫ab2πx[4x-x2-x] dx

V= ∫032π[3x2-x3] dx

V=2 π∫03[3x2-x3] dx

V=2π3x33-x4403

=2π 33-344-0

=2π27-814

=2π274

V=27π2

**Conclusion:**

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

V=27π2unit3