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 x axis. y=x³ y=0, x=2

Solution :

y = x³ => ³√y = x
or (y)^1/3 = x
or x = y^1/3

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

V = 2π_a∫^bp(y)h(y) dy
here p(y)=y^1/3, h(y)=y
when x = 2 and y = 3³ = 8
So a = 0 and b = 8
Plugging in all the values in the formula, we get

V = 2π₀∫⁸(y)^1/3.y dy

= 2π₀∫⁸(y)^4/3 dy (as 1/3 + 1/1 = (1+3)/3 = 4/3)

= 2π[y^7/3/(7/3)₀]⁸ (as 4/3 + 1/1 = (4+3)/3 = 7/3)

= 2π[(8)^7/3/(7/3)- (0)^7/3/(7/3)]

= 2π[((2)³)^7/3/(7/3)]

= 2π(2⁷/(7/3))

= 2π(128/(7/3))

= 2 * 3 * 128 * π / 7

= 768π /7


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