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Calculus/Calculating the volume of 2 functions rotated around the x-axis

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Question
Dear Abe,

I'm learning for my Calculus exam, but this exercise isn't very clear to me:

y=(x-2)^3
y=x-2

The formula for the volume is V=pi(integral |f(x)|dx

The correct result is 8pi/21

Would be delighted if you'd help me out with this exercise.

Regards,
Bob

Answer
Volume Problem
Volume Problem  
Hello Bob,

The two curves intersect at x=1, 2, and 3 (see attached graph).
However, it is not necessary to break the problem into two separate
integrals, since from x=1 to 2 and x=2 to 3, the outer radius of the
solid formed (by rotating about the x-axis) are both y-x-2 with y-(x-2)^3
being the inner radius.  Hence, one integral can take care of the entire
volume!  he integral also appears in the attached image (as wellas the
answer of 8pi/21.

I hope this helps.

Abe

Calculus

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Abe Mantell

Expertise

Hello, I am a college professor of mathematics and regularly teach all levels from elementary mathematics through differential equations, and would be happy to assist anyone with such questions!

Experience

Over 15 years teaching at the college level.

Organizations
NCTM, NYSMATYC, AMATYC, MAA, NYSUT, AFT.

Education/Credentials
B.S. in Mathematics from Rensselaer Polytechnic Institute
M.S. (and A.B.D.) in Applied Mathematics from SUNY @ Stony Brook

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