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=16π3unit3

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 x=f(y) about x- axis from a to b is

V= ∫ab2πy f(y)dy

where  0≤a≤b

2) Given:

The region bounded by x=1+y-22, x=2  rotated about the x- axis.

3) Calculation:

As the region is bounded by x=1+y-22, x=2 

Using the shell method, the typical approximating shell with radius y is

To find the height of the region, subtract the two given functions.

2-1+y-22=2-[1+y2-4y+4 ]

=-y2+4y-3

Therefore, the circumference is 2πy and the height is -y2+4y-3

To find the points of intersection,

2=1+y-22

2-1+y2-4y+4 =0

-y2+4y-3=0

y2-4y+3=0

y=1 and y=3

Therefore, a=1 and b=3

So, the total volume is

V= ∫ab2πy [-y2+4y-3 ] dy

V=∫132πy [-y2+4y-3]dy

V=2π∫13-y3+4y2-3ydy  

V=2π-y44+4y33-3y2213

=-2π 344-4·333+3·322-144-4·133+3·122

=-2π814-36+272-14-43+32

V=-2π-83

V=16π3

Conclusion:

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

V=16π3unit3