To be able to use the shell method, we consider a rectangular strip from the bounded plane region which 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)*

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

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:

`r = x`

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

`h =x^4/4-0`

`h=x^4/4`

For the boundary values, we have `x_1=0` to `x_2=4` .

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

`V = int_0^4 2pi* x*x^4/4*dx`

`V = int_0^4 (2pix^5)/4*dx`

`V = int_0^4 (pix^4)/2*dx`

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

`V = pi/2 int_0^4 x^5*dx`

Apply **power rule for integration**: `int x^n dy= x^(n+1)/(n+1)` .

`V = pi/2* x^(5+1)/(5+1)|_0^4`

`V = pi/2 *x^6/6|_0^4`

`V =(pix^6)/12|_0^4`

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

`V =(pi(4)^6)/12-(pi(0)^6)/12`

`V = (1024pi)/3 -0`

`V = (1024pi)/3` or `1072.33 ` (approximated value)