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 y axis. y=x³, y=0, x=2
Solution :

Volume of a Solid by rotating about y-axis is given by:

V = 2π_a∫^bp(x)h(x) dx
here p(x)=x³, h(x)=x
when y = 0, 0 = x³ or ³√0 = x or x = 0
So a = 0 and b = 2
Plugging in all the values in the formula, we get

V = 2π₀∫²(x)³.x dx

= 2π₀∫⁸(x)⁴ dx

= 2π[x⁵/5₀]²

= 2π[(2)⁵/5 - (0)⁵/5]

= 2π[32/5]

= 64π/5

So this how the volume of Solid of revolution is determined when the equations about the y axis.
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