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1、Section 10.3Riemann,Bernhard2Concepts of Triple IntegralsWe use triple integrals to find the volumes of three-dimensional shapes,the masses and moments of solids,and the average values of functionsof three variables.is a function defined on a closed bounded region(V)in space(,)f x y zIf the region o
2、ccupied by a solid ball.We number the cells that lie inside(V)from 1 to n in some order.We choose(,)kkkxyzin each cell and from a point 1(,).nnkkkkkSf xyzV 3Concepts of Triple IntegralsIf f is continuous and the bounding surface of(V)is made of smooth andkz,kx surfaces joined along continuous curves
3、,then asky approach zero independently,the sums Sn approach a limit 0()lim(,).ndVSf x y z dV The limit also existsWe call this limit the triple integral 三重积分三重积分 of f over(V).for some discontinuous functions.Element of volume4Concepts of Triple IntegralsIf for any partition of(V)and any selection of
4、 Pk,the limit of the sum exists.01()(,)lim(,).nkkkkdkVf x y z dVfV Then we say f is integrable over the domain(V),and the limit value is called the triple integral of the function f over the domain (V),denoted by 1(,).nkkkkkfV xyOzkV Element of volumeSuppose f is a function of three variables(),kVV
5、defined onin the subregion.Then,form the sum ,kkkkP ().kVis any point is the volume ofin space.()VThere is a regionthe subregion,Concepts of Triple IntegralsJust as the area of a plane region can be found by evaluate the doubleintegral,the volume of a region in space also can be found by evaluate Vo
6、lume().VdV 01()(,)lim(,).nkkkkdkVf x y z dVfV 2()V1()V6Properties of Triple Integralthen(),VSuppose(,)f x y zand(,)g x y zare both integrable over2.Additivity with respect to the domain of integration12()()()(,)(,)(,).VVVf x y z dVf x y z dVf x y z dV,where k is a constant.()()(,)(,)VVkf x y z dVkf
7、x y z dV(1)1.Linearity Propertyand12(),()VVSuppose that12()()()VVVhave no common part except for their Thenboundaries.(2)()()()(,)(,)(,)(,)VVVf x y zg x y z dVf x y z dVg x y z dV7Properties of Triple Integral4.Mean Value Theoremon().Vthen()()(,)(,)VVf x y z dVg x y z dV,if(,)(,)f x y zg x y z on().
8、V(1)()(,)0Vf x y z dV ,if(,)0f x y z (2)(,),(,)(),lf x y zLx y zV(4)If()(,).VlVf x y z dVLV3.Domination,such thatis a closed bounded,and connected()VSuppose that ,()V anddomain.()fC V Then there exists at least one point ()(,),.Vf x y z dVfV ()()(,)(,)VVf x y z dVf x y z dV(3)8How to Find Limits of
9、Integration in Triple IntegralsTo evaluate()(,)Vf x y z dVover a region(V),we integrate first with respect to z,then withrespect to y,finally with x.()(,)(,)Vf x y z dVdxdyf x y z dz 9How to Find Limits of Integration in Triple Integrals()(,)Vf x y z dVIntegrating first with respect to z,then with r
10、especty,finally with x,take the following steps.Step 1:A sketch.Sketch the region(V)along with its“shadow”()(vertical projection)in the xy plane.Label the upper and lower bounding surfaces of(V)and the upper and lower bounding curves of().10How to Find Limits of Integration in Triple IntegralsStep 2
11、:The z limits of integration.Draw a line M passing through a typical point(,)x yin()parallel to the z axis.As zincreases,M enters(V)at 1(,)zfx y These are the2(,).zfx y and leaves at z limits of integration.21(,)(,)()(,)zfx yzfx yf x y z dz d “First single and then double”11How to Find Limits of Int
12、egration in Triple IntegralsDraw a line L passing through a typical point(,)x yin()parallel to the y axis.As yincreases,L enters()at 1()ygx These are the2().ygx and leaves at y limits of integration.Step 3:The y limits of integration.12How to Find Limits of Integration in Triple IntegralsStep 4:The
13、x limits of integration.Choose x limits that include all linesthrough()parallel to the y axis.Then the integral is2211()(,)()(,)(,).x by gxzfx yx ay gxzfx yf x y z dzdydx“three single integrals”2211()(,)()(,)()(,)(,).bgxfx yagxfx yVf x y z dVf x y z dzdydx 13Computation of Triple IntegralsExample Ev
14、aluate (),VIxyzdV and0,x where the region(V)is enclosed by the planes0y 1.xyzy O z x 111(V)Oxy11()01zxyM()(,)|01,01x yxyx 0y Computation of Triple IntegralsSolution10()()xyVIxyzdVxyzdz d ()(,)|01,01,x yxyx 111000 xxyxyzdz dydx 1.720 Finish.15Computation of Triple Integrals2()zcczddz Solution The typ
15、ical cross-section can beThen shown as an ellipse.whereExample Evaluate 222222()(,)1,(,0).xyzVx y za b cabc2(),VIz dV 2()zccIz ddz Since 222222221()(,),|11zxyx ywhere zczzabcc 16Computation of Triple IntegralsSolution (continued)is()z then the area of 22()1,zzdabc Hence 34.15abc 2221cczIabz dzc Fini
16、sh.17Triple Integrals in Cylindrical CoordinatesDefinition Cylindrical CoordinatesCylindrical coordinates represent a point P as the(,)z in space by ordered triplesright figure.Equations Relating Rectangular(x,y,z)and Cylindrical(r,z)Coordinates222(,tan/).xyy xcos,sin,xyzzwhere0,02,.z 18The Volume E
17、lement in Cylindrical CoordinatesThe volume element for subdividing a region in space with cylindrical coordinatesis,dVd d dz and then triple integrals in cylindrical coordinatesare then evaluated as iterated integrals.()()(,)(cos,sin,)VVf x y z dVfzd d dz 19Finding a Volume in Cylindrical Coordinat
18、esExample Find the volume of the region(V)in space,which isand224.xyz bounded by 22zxySolution The equations of thesetwo surface in cylindrical coordinate are24z and2z It is easy to see that the projection region on xOy plane is 2()(,)2,02.or 22()(,)2,x y xy 20Finding a Volume in Cylindrical Coordin
19、atesSolution(continued)Then the volume can be found by Volume()VdV 22200(42)dd()Vd d dz 2222400dddz 4.2420222 Finish.21Triple Integrals in Spherical CoordinatesEquations Relating Spherical Coordinates to Cartesian andCylindrical Coordinates222220,0,02,().rrxyzzcoszr sinr cossin cosxrsinsin sinyrDefi
20、nition Spherical CoordinatesSpherical coordinates represent a point P as the(,)r in space by ordered triplesright figure.cossinsinsincosxryrzr22The Volume Element in Spherical CoordinatesThe volume element for subdividing a region in space with spherical coordinatesis2sin,dVrdrd d and the triple int
21、egrals in spherical coordinatesare then evaluated as iterated integrals.2()()(,)(sin cos,sin sin,cos)sinVVF x y z dVF rrrrdrd d ()(,)?Vf rdV How to find limits of integration in triple integrals2223Finding a Volume in Spherical CoordinatesExample Find the volume of the“ice cream cone”(V)cut from the
22、by the cone/3.solid sphere 1r Solution 2()sinVVrdrd d 2/312000sinrdrd d .3 Finish.24How to Integrate in Spherical Coordinatesover a region(V)in space in spherical()(,)Vf rdV To evaluatecoordinates,integrating first w.r.t.r,then w.r.t.and finally w.r.t.,take the following steps.Step 1:A sketch.over t
23、he xy plane.()its projection().VLabel the surfaces that boundalong withSketch the region()V2425How to Integrate in Spherical CoordinatesStep 2:The r limits of integration.Draw a ray M from the origin through()Vmaking an angle with the positivez axis.Also draw the projection of M on the xy plane(call
24、 the projectionL).The ray L makes an angle withwith the positive x axis.As r increases,M enters()Vand leaves2(,).rg at1(,)rg 2526How to Integrate in Spherical CoordinatesStep 4:The limits of integration.For any given ,the angle tomax.runs from min Step 3:The limits of integration.The ray L sweeps ov
25、er()as runs from a to b.Then,()(,)Vf rdV max2min1(,)2(,)(,)sin.ggf rrdrd d 2627Computation of Triple IntegralsExample(Page 263)Evaluate 2(),VIz dV where 22222222(,)|,()x y zxyzRxyzRR(V)RR/2Solution 1 Using cylindrical coordinatesSolution 2 Using spherical coordinatesSolution 3 Using the method of “f
26、irst single and then double”integration or“first double and then single”integration.28Computation of Triple IntegralsExample(Page 263)Evaluate 2(),VIz dV where 22222222(,)|,()x y zxyzRxyzRR(V)Solution 1 Using cylindrical coordinates29Computation of Triple IntegralsSolution 2 Using spherical coordinates30Computation of Triple IntegralsSolution 3 Using the method of “first double and then single”integration.