702 Iterated and multiple integrals
where S is the cube having four of its vertices at the points (0,0,0), (a,0,0),
(0,a,0), (0,0,a). .Ins.:
'(1 + 3a)2 log (1 + 3a) - 3(1 + 2a)2 log (1 + 2a) + (1 + a)2 log (1 + a).
(^14) Supposing that a, b, c are positii a numbers, evaluate
IJJ?sin{7r (a+b +c/1 dT'
where T is the tetrahedron having vertices at the points (0,0,0), (a,0,0), (0,b,0),
(0,0,c). .Ins.: (72 - 4)abc/2a3.
13.7 Triple integrals; cylindrical coordinates In some cases a
solid S (or set S in E3 having positive volume) and a function f defined
over S are such that the triple integral defined as in (13.624) by the
formula(13.71) M,f (P) dS = lim I f (P) iS
can advantageously be expressed in terms of cylindrical coordinates p,
0, z. When we use cylindrical coordinates, the set S is partitioned into
subsets S1, S2, ' ' ' ,S. by planes through the z axis having cylindrical
equations 0 _ 4o, 0 = 01, ' '' , 0 _ 0,,by circular cylinders having
cylindrical equations p = po, p = pi, '. , p = p,,',and by planes parallel
to the xy plane having the cylindrical equations z = zo, z = z1, ,
z = z.,,. Figure 13.72 shows a typical subset containing a point havingFigure 13.72cylindrical coordinates p, 0, z. This subset has height Oz and, as we
learned when studying polar coordinates, its base has area exactly or
approximately equal to p Ac Isp. Thus we use the formula
(13.73) AS = p AO Ap Oz