Full text: Advanced calculus

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,
	        
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