Home > Calculus > Determining the Volume of a Solid of Revolution

How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #27y=x^3#, y=0 , x=6 revolved about the y=8?

Answer 1

Scarlett Allison

The volume is #=(960pi)/7#

The volume of a small shell

#dV=pi(8^2-(8-y)^2)dx#

As, #y=x^3/27#

#dV=pi(64-(8-x^3/27)^2)dx#

#dV=pi(64-64+16x^3/27-x^6/729)dx#

#V=piint_0^6(16x^3/27-x^6/729)dx#

#=pi[16x^4/4*1/27-x^7/7*1/729]_0^6#

#=pi(4*6^6/27-6^7/(7*729)-0)#

#=pi(5184/27-384/7)#

#V=(960pi)/7#

graph{(y-x^3/27)(y-8)=0 [-19.15, 16.9, -4.32, 13.7]}

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.