Using Shell or Disc Method to Find Volume of the Solid

Disc method and Shell(cylinder) method of integration are the two different methods of finding volume of solid of a revolution, using rectangular coordination system the functions are defined in terms of x in the below problem.

Topic : Disc or Cylinder Method of Finding Volume of the Sphere.
Problem : Use the disc or shell method to find the volume of the solid generated by revolving the regions bounded by the graphs of the equations about the specified line y = 8.
y=x³ y=0, x=2
Solution :
Here the solid is rotated along x-axis
y = x³ => x = (y)^1/3

or x = y^1/3

when y = 8, x = (8)^1/3 = (2³)^1/3 = 2^3*1/3 = 2

So a = 0, b = 2

Volume of a Solid by rotating about y = 8 is given by:

V = 2π₀∫²(y)^1/3 . y dy

= 2π₀∫²(y)^4/3 dy

= 2π[y^7/3/(7/3)₀]²

= 2π[(2)^7/3/(7/3)- (0)^7/3/(7/3)]

= 2π[((2⁷)^1/3/(7/3)]

= 2π ³√(128)/(7/3)

= 6.³√(128)π/7

= 4.32π