How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #y=3x^4#, y=0, x=2 revolved about the x=4?

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Answer 1 · 2016-07-20T17:03:30

#(128/15)pi# cubic units

The volume of this annular solid, open at y = 0 and y = 48 and

appearing as an inverted funnel is

#pi int(4^2-(x-4)^2) d y#, from y=0 to y = 48.

#=pi int (2 x - x^2) d y#, from y=0 to y = 48..

#=pi int (2 (y/3)^(1/4) - (y/3)^(1/2) )d y#, from y=0 to y = 48..

#=pi [2(y/3)^(5/4)/(5/4) - (y/3)^(3/2)/(3/2)]#. between y = 0 and y = 48

#=.pi [2(48/3)^(5/4)/(5/4) - (48/3)^(3/2)/(3/2)]#

#=.pi [256/5 - 128/3)]#

#=(128/15)pi# cubic units.