Advanced Calculus fi..

232 Advanced Calculus, Fifth Edition , . + 34.4 TRIPLE INTEGRALS AND MULTIPLE INTEGRALS IN GENERALThe notion of double integral generalizes to integrals of functions of three, four, ormore variables:These are called tGple, quadruple, . . . integrals. In general, one terms them multipleintegrals, the double integral being the simplest multiple integral.For the triple integral, for example, one considers a function f (x, y, z) definedin a bounded closed region R of space. One subdivides R into rectangular parallelepipedsby planes parallel to the coordinate planes, numbers the parallelepipedsinside R from 1 to n, and denotes the ith volume by Ai V. The triple integral is thenobtained as the limit of a sum:ahere h is the mesh, the maximum diagonal of the AiV. The point (xf, yf, zf) isarbitrarily chosen in the ith parallelepiped. The existence of a unique limit can beshown if f (x, y, z) is continuous in R.The simplest theory holds for a region R described by inequalities such as thefollowing:For this region, one can reduce the triple integral to an iterated integral by the equationas in the theorem of Section 4.3.EXAMPLE If R is described by the inequalities05x51, 05y5x2, o ~ z ~ X + yand f = 2x - y - z, one has3The limits of integration can be determined by the following procedure. Letthe order chosen be dz dy dx as in the preceding example. Then we determine the