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Math S21a: Multivariable calculusOliver Knill, Summer 2013Lecture 17: Triple integralsIff(x, y, z) is a function of three variables andEis asolid regionin space, thenintegraltext integraltext integraltextEf(x, y, z)dxdydzis defined as then→ ∞limit of the Riemann sum1n3summationdisplay(i/n,j/n,k/n)∈Ef(in,jn,kn).As in two dimensions, triple integrals can be evaluated by iterated 1D integral computations. Hereis a simple example:1AssumeEis the box [0,1]×[0,1]×[0,1] andf(x, y, z) = 24x2y3z.integraltext10integraltext10integraltext1024x2y3z dzdydx.To compute the integral we start from the coreintegraltext1024x2y3z dz= 12x3y3, then integrate themiddle layer,integraltext1012x3y3dy= 3x2and finally and finally handle the outer layer:integraltext103x2dx=1.When we calculate the most inner integral, we fixxandy. The integral is integrating upf(x, y, z) along a line intersected with the body. After completing the middle integral, wehave computed the integral on the planez=constintersected withR.The most outerintegral sums up all these two dimensional sections.There are two important reductions to compute volume and triple integrals: the”washermethod”and the ”sandwich method”. The washer method from single variable calculusreduces the problem directly to a one dimensional integral.The new sandwich methodreduces the problem to a two dimensional integration problem.Thewashermethodslices the solid along a line. Ifg(z) is the double integral alongthe two dimensional slice, thenintegraltextbag(z)dz. Thesandwich methodsees the solidsandwiched between the graphs of two functionsg(x, y) andh(x, y) over a commontwo dimensional regionR. The integral becomesintegraltext integraltextR[integraltexth(x,y)g(x,y)f(x, y, z)dz]dA.2An important special case of the sandwich method is the volumeintegraldisplayRintegraldisplayf(x,y)01dzdxdy .below the graph of a functionf(x, y) and above a regionR, considered part of thexyplane.It is the integralintegraltext integraltextRf(x, y)dA. We actually have computed is a triple integral. Think ofvolume as a triple integral from now on.1
3Find the volume of the unit sphere.Solution:The sphere is sandwiched between the graphsof two functions. LetRbe the unit disc in thexyplane. If we use thesandwich method,we getV=integraldisplay integraldisplayR[integraldisplay√1−x2−y2−√1−x2−y21dz]dA .which gives a double integralintegraltext integraltextR2√1-x2-y2dAwhich is of course best solved in polarcoordinates. We haveintegraltext2π0integraltext10√1-r2r drdθ= 4π/3.With thewasher methodwhich is in this case also calleddisc method, we slice alongthezaxes and get a disc of radius√1-z2with areaπ(1-z2). This is a method suitablefor single variable calculus because we get directlyintegraltext1−1π(1-z2)dz= 4π/3.