Mathematical Methods for Physics and Engineering - Matematica.NET

6.3 APPLICATIONS OF MULTIPLE INTEGRALSand let N →∞as each of the volumes ∆V p → 0. If the sum S tends to a uniquelimit, I, then this is called the triple integral of f(x, y, z) over the region R and iswritten∫I = f(x, y, z) dV , (6.5)Rwhere dV stands for the element of volume. By choosing the subregions to besmall cuboids, each of volume ∆V =∆x∆y∆z, and proceeding to the limit, wecanalsowritetheintegralas∫∫∫I = f(x, y, z) dx dy dz, (6.6)Rwhere we have written out the element of volume explicitly as the product of thethree coordinate differentials. Extending the notation used for double integrals,we may write triple integrals as three iterated integrals, for example,I =∫ x2x 1dx∫ y2(x)y 1(x)dy∫ z2(x,y)z 1(x,y)dz f(x, y, z),where the limits on each of the integrals describe the values that x, y and z takeon the boundary of the region R. As for double integrals, in most cases the orderof integration does not affect the value of the integral.We can extend these ideas to define multiple integrals of higher dimensionalityin a similar way.6.3 Applications of multiple integralsMultiple integrals have many uses in the physical sciences, since there are numerousphysical quantities which can be written in terms of them. We now discuss afew of the more common examples.6.3.1 Areas and volumesMultiple integrals are often used in finding areas and volumes. For example, theintegral∫ ∫∫A = dA = dx dyRRis simply equal to the area of the region R. Similarly, if we consider the surfacez = f(x, y) in three-dimensional Cartesian coordinates then the volume under thissurface that stands vertically above the region R is given by the integral∫ ∫∫V = zdA= f(x, y) dx dy,Rwhere volumes above the xy-plane are counted as positive, and those below asnegative.191R