`y = x^2 , y = 4x-x^2` Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-axis.

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To be able to use the shell method, a rectangular strip from the bounded plane region should be parallel to the axis of revolution. By revolving multiple rectangular strip, it forms infinite numbers of this hollow pipes or representative cylinder.

 In this method, we follow the formula: `V = int_a^b` (length * height * thickness)

or `V = int_a^b 2pi` * radius*height*thickness


radius (r)= distance of the rectangular strip to the axis of revolution

height (h) = length of the rectangular strip

thickness = width  of the rectangular strip  as dx or dy.

For the bounded region, as shown on the attached image, the rectangular strip is parallel to y-axis (axis of rotation). We can a let:


`h=f(x)` or `h=y_(above)-y_(below)`

`h=(4x-x^2)-x^2 = 4x-2x^2`

thickness `= dx`  

 Boundary values of x from `a=0 ` to `b =2` .

Plug-in the values on `V = int_a^b` `2pi` * radius*height*thickness, we get:

`V = int_0^2 2pi*x*(4x-2x^2)*dx`

Apply Law of Exponent:` x^n*x^m = x^((n+m))` .

`V = int_0^2 2pi(4x^2-2x^3)dx`

Apply basic integration property: `int c*f(x) dx = c int f(x) dx`

`V = 2pi[ int_0^2 (4x^2-2x^3)dx]`

Apply basic integration property:`int (u-v)dx = int (u)dx-int (v)dx.`

`V = 2pi[ int_0^2 (4x^2) dx-int_0^2 (2x^3)dx]`

Apply Power rule for integration: `int x^n dx= x^(n+1)/(n+1).`

`V = 2pi[ (4*x^((2+1))/((2+1)))-(2 *x^((3+1))/((3+1)))]|_0^2`

`V = 2pi[ 4*x^3/3-2*x^4/4]|_0^2`

`V = 2pi[ (4x^3)/3-x^4/2]|_0^2`

Apply definite integration formula: `int_a^b f(y) dy= F(b)-F(a).`


`V = 2pi[ (4(2)^3)/3-(2)^4/2]-pi[ (4(0)^3)/3-(0)^4/2] `

`V = 2pi[ 32/3-8]-pi[ 0-0]`

`V = 2pi[ 8/3]-0`

`V = (16pi)/3` or `16.76` (approximated value)

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