Find the volume of the solid by using cylindrical shells?

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Answer 1 · 2018-05-28T06:04:36

#color(blue)[V=2piint_0^1(y)(sqrty-y^3)*dy=(2*pi)/5 (unite)^3]#

the volume by cylindrical shell method when the curve rotating about the #"x-axis"# is given by:

#color(red)[V=2piint_a^b(y)(x_2-x_1)*dy]#

#x_2=+-sqrty#
but the area enclosed by the curve lies in the first quadrant.

so #x_2=sqrty#

#x_1=y^3#

lets find the interval of integral.

#sqrty=y^3 # square both sides

#y^6=y rArr y^6-y=0#

#y*(y^5-1)=0#

#y=0 and y=1#

the interval of the integral #y in [0,1]#

now let set up the integral:

#V=2piint_0^1(y)(sqrty-y^3)*dy#

#V=2piint_0^1(y^(3/2)-y^4)*dy=[-(2*pi*(y^5-2*y^(5/2)))/5]_0^1#

#=(2*pi)/5 (unite)^3#

show below the region revolving (shaded region) :

James

show the link below that will help you to understand how to find the volume by cylindrical shell method:
cylindrical shell method