98
CALCULUS
9. Develop the iterated integral by first holding a single coordi
nate fast (e.g. set r = r t ) and then obtaining a double integral.
10. Write out the twelve iterated integrals, — six, as three-fold
simple integrals, and six as a double integral combined with a simple
integral.
11. Apply to Exercise 5 a sufficient number of each type of the
iterated integrals considered in the preceding problem to make sure
that you understand the rest.
12. A tetrahedron has its faces in the coordinate planes x = 0,
y = 0, and the planes
z = x -(- y, z = a.
Express as an iterated integral of the type (14) the volume integral
(2), determining explicitly the limits of integration.
13. The density of a cube is proportional to the distance from its
centre. Eind its mass.
14. Compute the moment of inertia of the cube of the preceding
problem about an axis through the centre parallel to four of the edges.
4. Conclusion; Cylindrical Coordinates. The cylindrical coordi
nates of a point are defined as in the accompanying figure.* They
z are a combination of polar coordinates in the
(x, y)-plane and the Cartesian z.
—:—7^ x = r cos 9, y = r sin 9, z = z.
/ The element of volume is shown in Fig. 32. The
Fig 31 n
lengths of the edges adjacent to P, — they meet
at right angles there, — are: Ar, rA0, Az. Hence the volume, AT 7 ,
of the element differs from r Ar A9 Az by an infinitesimal of higher
order, and we have :
Z
From Duhamel’s Theorem it follows, then,
that in taking the limit of the sum (1), § 1,
AT 7 * may be replaced by r t ArA0Az, and so,
Fig. 32
setting /(ar, y, z)= F(r, 9, z),
we obtain:
* Analytic Geometry, p. 587,