UNIT 13 VOLUME INTEGRAL Structure 13.1 Introduction 0bjectives 13.2 Triple Integral 13.2.1 Definition 13.2.2 Pmpenies of Triple Integrals 13.2.3 Volume 13.2.4 Evaluation of Triple integrals 13.2.5 Physical Applications of Triple Integrals 13.3 Transformation of Volume Integrals into Surface Integrals 13.3.1 Gauss Divergence Theorem 13.3.2 Consequen~sand Applications of Divergence Theorem 13.3.3 Integral Definitionsof Gradient, Divergence and Curl 13.3.4 Physical Interpretation of Divergence Theorem 13.3.5 Modelling of Heat Flow 13.3.6 Green's Theorem and Green's Formula 13.3.7 A Basic pmperty of Solutions of laplace's Equation 13.4 Irrotational and Solenoidal Vecto. Fields Revisited 13.5 Sununary 13.6 Solutions/Answers 13.1 INTRODUCTION In the previous unit, you have studied about liile integrals, double ilitegrals a~idsurface integrals. In the process you have learnt to transform double integrals and surface integral into line integrals. You had learnt that line integral is the generalization of a single integral and a surface integral is a sort of generalization of a double integral. In this unit, we shall give another generalization of double integral called triple integrals or volume integrals. We shall first of all define triple integrals in Section 13.2, wherein we shall also give the properties and evaluation of such integrals. This section will be closed with some physical applications of triple integrals. We had made use of integral transform theorems - Green's Theorem and Stoke's Theorem - in the last unit. In this unit, we shall discuss another important intergral transform theorem, known as Gauss Divergence Theorem, which helps in the transformation of volume integral to surface intergal and conversely. Divergence Theorem has many important consequences and various applications, some of which have been discussed in Section 13.3. We had earlier discussed solenoidal vector fields al~dimtatiollal vector fields in Unit 11. With our knowledge of vector calculus, we have revisited these concepts in Section 13.4, where we have now given integral form conditions for vector fields to be solenoidal and irrotational. The summary of the results discussed in this unit is presented at the end of this unit. Objectives After going through this unit, you should be able to + define a triple integral, * state properties of triple integrals, + evaluate triple integrals as repeated integrals, * give some physical applications of triple integrals, * lean1 the method and conditions, under which a volume integral can be transformed into a surface integral and conversely (Gauss Divergence Ttieorem), * apply divergence theorem for the evaluation of surface integrals/volume integrals, * define gradient, divergence and curl in tenns of integrals, * understand the physical interpretation of divergence theroem, Volome Integral * use divergence theorem to set up heat equation, * derive Green's theorem in space and Green's formula, using divergence theorem, * lean1 the uniqueness property of solutions of Laplace Equation, and * express conditions for a vector field to be solenoidal and irrotation in terms of integrals. 13.2 TRIPLE INTEGRALS In Section 12.1, the concept of a line integral was introduced. Section 12.2 was devoted to the properties and evaluation of double integrals, in which the integrand is a function of two variables. A natural generalization of double integrals was provided in Section 12.5 in terms of surface integrals, where though the integrand may be a function of three variable, but it is difined on a surface only and it is still evaluated as a double integral. However, there are many physical and geometrical situations, where the integrand may be a fuilction defined in a region in space and itegration may have to be carried out with respect to all the three variables involved. This gives rise to triple integrals. A triple integral is again a generalization of double integral (introduced in Section 12.2). 13.2.1 Definition For defining a triple integral, consider a function f (x, y, z) defined on a bounded region R in space (such as a solid cube, a ball, a truncated cone or the space between two concentric spheres). We subdivide the region R into rectangular cells by planes parallel to the three coordinate plai~es.The parallelopiped cells may have the dimensions of Ax, Ay and A z. We number the cells inside R in same order AV,,AV, ,..., AV, In each such parallelopiped cell; we choose an arbitrary point, say, (xbyb zk) in the kth Figu~13.1:Partitioningasolid with rectangular cells of volume AV. parallelopiped cell, and form the sum n s~= 1 f (xb Yk, zk)AVk k- 1 (13.1) where AVk is the volume of kth cell. We now arbitrarily do this for larger and larger positive integer 11 so that the lei~gthsof the edges of the largest parallelopiped of subdivision approaches zero as n approaches infinity. In this way, we obtain a sequence of real numbers S,, ,Snz,.... If we assume that I(x,y, z) is coi~tii~uousin a domain containing R and R is bounded by finitely many smooth curves, then as Ax, Ay, A z all approach zero with n approaching infinity, the sum S, will approach a limit [independent of choice of subdivisions and corresponding points (xb Yb zk)] Lt s, =$$sf(4 Y, Z) dv, (13.2) n-rm R which is called the triple integral off (x, y, z) over the region R. It may be mentioned that the liinit in equation (13.2) may exist for some discoiitinuous functions also. 13.2.2 Properties of Triple Integrals Triple integrals share many algebric properties with double and single integrals. Writing F for F (x, y, z) and G for G (x, y, z), some of the properties of triple integrals (i) $$$K F N- K$$F N (any constant K) R R (ii) $$$(F * G) N =$$SF ~v*$$$G dv R R R (iii) $$$FNZO if FaOinR R (iv) $$$FN~$$$G~v if FZG~~R R R The triple integrals have the above properties because the sums that approxiinate the111 have these properties. Triple integrals also have a domain additivity property that proves useful in physics, mathematics as well as in engineering. If the domain R of a coi~tinuousfunction F be partitioned by smooth surfaces into a finite number of regions R1, R2, .. Rn in space, then $$$FN=$$$F ~v+$$JF dv+ ... +$$$F~v R R 1 R2 R" 13.2.3 Valume Iff (x, y, z) = 1 is the constant function whose value is one, then the sums in Equation (1) reduce to n n s,,=c teAvk=cAvk k- 1 k- 1 As Ax, Ay and A z all approach zero, the cells A Vk become smaller and smaller and more numerous and fill up more of R. We, therefore, define the volume of R to the triple integral of constant function f (x, y, z) = 1 over R. Hence, n Volume O~R= ~t C A vx =$$$N n-w k-1 R The triple integral is seldom evaluated directly froin its defiilitioii as the liinit of a sum. Instead, one applies a three-dimensional version to evaluate the integral by repeated single integrations. This method is explained below. 13.2.4 Evaluation of Triple Integrals To express the triple integral as a repeated integral we divide the region R into eleineiitary cuboids by planes parallel to the three coordinate planes. The volume R may then be considered as the sum of vertical colun~nsextending from the lower surface of R, say z = z~ (x, y) to its upper surface z = z2 (x, y). The bases of these columns are the elementry area bA, which cover a certain region Ain the xy- plane when all the columns in R are taken. P I X Figure 13.2 : Evaluation of triple integral Thus, if we take the sum over the elementary cuboids in the same vertical column first, and then take the sum for all the columns in R, we can write equation (13.1) as Volume ImtqpL t where (xbyb zk) is a point in the phcuboid above the rthelementary area Mr. Taking the limit when the dimensions of 6Ar and A z tend to zero, the above sum yields Now integration with respect to z is performed fint, keeping x and y constant. .The remaining integration is performed as for the double integral. Therefore, if A is bounded by the curves then triple integral in equation (13.5) may be written as where the three integrations are performed in order from the innermost to the outermost. It should, however, be noted that the region A is the projection in the xy - plane of the bounding surface of the volume bounded by R. In case the lateral surface of the system reduces to zero, as in Figure 13.3, we may find the boundary of A by eliminating z between the two equations z = fl(x, y) and z = f2(x, y). Figure 13.3 : Lateral surface reduces to zero I This gives an equation that contains no z and that defines the boundary of A in the xy-plane. To obtain the z-limits of integration in any particular instance, we may use a procedures as follows : We imagine a line L through a point (x, y) in area A and parellel to z-axis. As z increase, the line enters R at z - fi(x, y) and leaves R at z I&, y). These give the lower and upper limits of the integration with respect to z. The result of this integration is a function of x and y alone, which we can integrate supplying the limits as in double integrals.