`y = 4x - x^2, y = 3` Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves...

`y=4x-x^2, y=3`

The point of intersection of the curves will be ,

`4x-x^2=3`

`4x-x^2-3=0`

`-x^2+4x-3=0`

`-(x^2-4x+3)=0` 

`x^2-4x+3=0`

factorizing the above equation,

`(x-3)(x-1)=0`

x=3 , x=1

The shell has radius (x-1) , circumference `2pi(x-1)`  and height `(4x-x^2)-3`

Volume generated by rotating the region bounded by the given curves about x=1 (V)=`int_1^3(2pi)(x-1)(4x-x^2-3)dx`

`V=int_1^3(2pi)(4x^2-x^3-3x-4x+x^2+3)dx`

`V=(2pi)int_1^3(-x^3+5x^2-7x+3)dx`

`V=2pi[-x^4/4+5x^3/3-7x^2/2+3x]_1^3`

`V=2pi[-3^4/4+5/3*3^3-7/2*3^2+3*3]-2pi[-1^4/4+5/3*1^3-7/2*1^2+3*1]`

`V=2pi((-81/4+45-63/2+9)-(-1/4+5/3-7/2+3))`

`V=2pi(9/4-11/12)`

`V=2pi(16/12)`

`V=(8pi)/3`

`y=4x-x^2, y=3`

The point of intersection of the curves will be ,

`4x-x^2=3`

`4x-x^2-3=0`

`-x^2+4x-3=0`

`-(x^2-4x+3)=0` 

`x^2-4x+3=0`

factorizing the above equation,

`(x-3)(x-1)=0`

x=3 , x=1

The shell has radius (x-1) , circumference `2pi(x-1)`  and height `(4x-x^2)-3`

Volume generated by rotating the region bounded by the given curves about x=1 (V)=`int_1^3(2pi)(x-1)(4x-x^2-3)dx`

`V=int_1^3(2pi)(4x^2-x^3-3x-4x+x^2+3)dx`

`V=(2pi)int_1^3(-x^3+5x^2-7x+3)dx`

`V=2pi[-x^4/4+5x^3/3-7x^2/2+3x]_1^3`

`V=2pi[-3^4/4+5/3*3^3-7/2*3^2+3*3]-2pi[-1^4/4+5/3*1^3-7/2*1^2+3*1]`

`V=2pi((-81/4+45-63/2+9)-(-1/4+5/3-7/2+3))`

`V=2pi(9/4-11/12)`

`V=2pi(16/12)`

`V=(8pi)/3`