Calculus: Analytic Geometry and Calculus, with Vectors

13.7 Triple integrals; cylindrical coordinates 703

to put (13.71) in the form

(13.74) fff3f(p,O,z)p do dp dz= limIf(P,o,z)P Li4 Op Oz.

Assuming that f is bounded over S and that f is sufficiently continuous to
make all of the integrals exist, we can use Theorem 13.63 to express this
triple integral as an iterated integral. When limits of integration for
iterated integrals are being determined, information obtained by looking
at Figure 13.72 can be helpful. Adding subsets for which z varies (p
and 0 being fixed) yields a vertical column. Adding columns for which
p varies (0 being fixed) yields a whole or a part of a wedge which in some
cases looks like a conventional wedge of a cake or orange or lemon.
Adding the wedges obtained for appropriate values of f then gives the
entire solid S. Results of performing summations and integrations in
different orders are easily described. For example, adding subsets for
which 0 varies (a and p being fixed) yields all or part of a circular hoop
or ring, and there are two ways in which these hoops can be added to
yield more extensive parts of S.
Supposing that S is a right circular cylindrical solid bounded by the
graphs of the equations p = R, z = 0, and z = H and that the density
(mass per unit volume) at the point having cylindrical coordinates
(p,4,z) is 5(p,o,z), we set up an integral for the polar moment of inertia
M='o.Z=o of S about the x axis. For the volume AS of a subset of the

Figure 13.75

solid, which is shown in Figure 13.75, we use the formula

(13.76) AS = pLOLipAZ.

To get the mass AM of the subset, we multiply by the density (mass per

unit volume) 5(p,o,z) to obtain

(13.761) AM = 5(p,4,z)p Li4 Op Oz.

Then we must realize what we are trying to do and multiplythis by

(13.762) z' + (p sin ¢)=,