y = x^4/4 , y=0 , x=4 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.

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)