Mathematical Methods for Physics and Engineering - Matematica.NET

MULTIPLE INTEGRALSzcdV = dx dy dzdzdxdybyaxFigure 6.3 The tetrahedron bounded by the coordinate surfaces and theplane x/a + y/b + z/c = 1 is divided up into vertical slabs, the slabs intocolumns and the columns into small boxes.◮Find the volume of the tetrahedron bounded by the three coordinate surfaces x =0, y =0and z =0and the plane x/a + y/b + z/c =1.Referring to figure 6.3, the elemental volume of the shaded region is given by dV = zdxdy,and we must integrate over the triangular region R in the xy-plane whose sides are x =0,y =0andy = b − bx/a. The total volume of the tetrahedron is therefore given by∫∫V = zdxdy=R= c= c∫ adx0∫ a0∫ a0∫ b−bx/a0(dy c 1 − y b − x )adx[y − y22b − xy a( bx2dx2a − bx 2 a + b 2] y=b−bx/ay=0)= abc6 . ◭Alternatively, we can write the volume of a three-dimensional region R as∫ ∫∫∫V = dV = dx dy dz, (6.7)RRwhere the only difficulty occurs in setting the correct limits on each of theintegrals. For the above example, writing the volume in this way corresponds todividing the tetrahedron into elemental boxes of volume dx dy dz (as shown infigure 6.3); integration over z then adds up the boxes to form the shaded columnin the figure. The limits of integration are z =0toz = c ( 1 − y/b − x/a ) ,and192