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

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)|²dx

The correct result is 8pi/21

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

Regards,
Bob

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

I hope this helps.

Abe

Calculus

Volunteer

#### 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