Page 202 - How To Solve Word Problems In Calculus

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V shell = V outer − V inner
cylinder cylinder

2
2
= πr h − πr h
2
1

2
= π r − r 1 2 h
2
= π(r 2 + r 1 )(r 2 − r 1 )h

r 2 + r 1
= 2π h(r 2 − r 1 )
2

r 2 + r 1
If we let r av = represent the average radius and
2
r = r 2 − r 1 represent the shell wall thickness, we may write

V shell = 2πr av h r

As r → 0, and the number of shells within the solid →∞,
the sum of their volumes will approach the volume of the solid
of revolution.
As a mnemonic device we may represent the average ra-
dius by r, the length of the shell by h, and the (infinitesimal)
shell wall thickness by dr. The volume of a typical shell may
be thought of as 2πrhdr and the total volume is

b

V = 2π rhdr
a

In a given problem, dr will be replaced by either dx or dy, de-
pending upon the axis of rotation (dx if rotated about the
y axis and dy if rotated about the x axis). In either case,
the height h and the radius r must be expressed in terms of
the variable of integration. The limits of integration must cor-
respond to the variable of integration as well.

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