#### To determine

**To find:**

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

#### Answer

V=16π unit3

#### Explanation

**1) Concept:**

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

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

V= ∫ab2πyf(y)dy

where, 0≤a≤b

**2) Given:**

The region bounded by x=-3y2+12y-9, x=0 rotated about the x- axis.

**3) Calculation:**

As the region is bounded by x=-3y2+12y-9, x=0 using shell method, find the typical approximating shell with the radius y.

Therefore, the circumference is 2πy and the height is x=-3y2+12y-9

a=1 and b=3

So, the total volume is

V= ∫ab2πy [-3y2+12y-9]dy

V= ∫132π [-3y3+12y2-9y ] dy

V=2π∫13-3y3+12y2-9ydy

V=-6π∫13y3-4y2+3ydy

V=-6πy44-4y33+3y2213

=-6π 344-4·333+3·322-144-4·133+3·122

=-6π814-36+272-14-43+32

V=-6π-83

V=16π

**Conclusion:**

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

V=16π unit3