To be able to use the shell method, the 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 hollow pipes or representative cylinder.

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

This can be also expressed as:

`V=int_a^b (2pi*` * radius * height *thickness of rectangular strip)*

or `V=int_a^b (2pir *` *length of rectangular strip *thickness of rectangular strip)* where r = distance of the rectangular strip from the axis of rotation.

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

`radius=y`

`height=y -(y^2-2) = y-y^2+2`

Note: When we squared both sides of `y =sqrt(x+2)` , we get:` y^2 =x+2.`

It can be rearranged into `x= y^2-2.`

thickness`=dy`

the boundary values of y is `a=0` to` b=2`

Then the integral set-up will be:

`V= int_0^2 2 pi *y * (y-y^2+2) *dy`

Simplify:`V= int_0^2 2 pi ( y^2-y^3+2y)dy`

Apply the basic integration property: `int c f(x) dx - c int f(x) dx`

`V= 2pi int_0^2 ( y^2-y^3+2y)dy`

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

`V= 2pi [int_0^2 ( y^2) dy-int_0^2(y^3)dy+int_0^2(2y)dy]`

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

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

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

Apply the definite integral formula:` int_a^b f(x) dx = F(b) - F(a)` , we get:

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

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

`V= (16pi)/3 -0`

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

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