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 cylinders.

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

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

where:

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 let:

`r=x`

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

`h =4-(4x-x^2) = 4-4x+x^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 pi*x*(4 -4x+x^2)*dx`

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

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

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

Apply basic integration property:`int (u+-v+-w)dy = int (u)dy+-int (v)dy+-int(w)dy` to be able to integrate them separately using Power rule for integration: `int x^n dx = x^(n+1)/(n+1).`

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

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

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

Apply the definite integral formula: `int _a^b f(x) dx = F(b) - F(a)` .

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

`V = 2pi[ 8-32/3+4] - 2pi[0-0+0]`

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

`V = (8pi)/3` or `8.38` (approximated value)

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