微积分全英微积分全英 (75).pdf

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1、Exercises Lesson（III）for Triple integral,line and surface integralExample 1Show that if is a positive integer,then 1lim0.pnn=2Let (,)|1,0,0,0,calculate.x y zxyzxyzy dS=+=(,)|1,0,0Dx yxyxy=+223Dy dSydxdy=112003312ydyy dx=Example 2Show that if is a positive integer,then 1lim0.pnn=2233 is the upper sid

2、e of the surface (1)calculate(1)(1)(1)zxyzIxdydzydzdxzdxdy=+=+，221Let is the lower side of surface1,1xyz+=1Let the area enclosed by and is.133(1)(1)(1)xdydzydzdxzdxdy+223(1)3(1)1xydzdxdy=+Example 2Show that if is a positive integer,then 1lim0.pnn=2233 is the upper side of the surface (1)calculate(1)

3、(1)(1)zxyzIxdydzydzdxzdxdy=+=+，1331,0(1)(1)(1)0zdzxdydzydzdxzdxdy=+=0 xdxdydzydxdydz=221122200(337)(37)4rIxydxdydzddrrrdz=+=+=Example 3Show that if is a positive integer,then 1lim0.pnn=222222 2,starting point (0,2,0),ending point:(0,2,0)calculate()()LL zxyzxABIyz dxzxy dyx y dz=+:：1Let straight line

4、 from A to BL：，1The projection curves of and on the plane areLLxOy221:22(0),clockwise;:0,:22lxyxlxy+=1The area enclosed by and is,llDzx=Example 3Show that if is a positive integer,then 1lim0.pnn=222222 2,starting point (0,2,0),ending point:(0,2,0)calculate()()LL zxyzxABIyz dxzxy dyx y dz=+:：2222()()

5、LIyz dxzxy dyx y dz=+111222222 ()()L LLllyx dxxxy dyx y dxydy+=+122 ()llyxx ydxydy+=+22 (12)12DDx y dxdydxdy=+=Example 4Show that if is a positive integer,then 1lim0.pnn=222 is the intersection of sphere 1 and plane 0LLxyzxyzxyds+=+=22210Lxyzxyz+=+=：，,have rotation symmetryx y z1()3LLLLxydsxzdsyzdsy

6、zxyxz ds=+222211()()32Lyxzxyzds=+211012663Lds=Example 5Show that if is a positive integer,then 1lim0.pnn=2222calculate(),is the area bounded by surfaces and 1.xz dvzxyzxy+=+=212244000011cossin2sin248|zdvddrrdr=()8xz dv+=is symmetric about the plane0yozxdv=Example 6Show that if is a positive integer,

7、then 1lim0.pnn=222Calculate(),is a three-dimensional region formed 2by a surface rotating around the-axis and 4.0 xyz dvyzzzx+=422222000()()zxyz dvdzdrz rdr+=+42025643z dz=Example 7Show that if is a positive integer,then 1lim0.pnn=2222Calculate,(,)|1z dvx y zxyz=+From symmetry of functional displace

8、ment,21220001sin3ddrrdr=22221()3z dvxyzdv=+114223515=Example 8Show that if is a positive integer,then 1lim0.pnn=2Curve:(02),calculateLL yxxxds=2220114(14)8x dx=+2201(2)Lxdsxxdx=+32220113(14)126|x=+=Example 9Show that if is a positive integer,then 1lim0.pnn=22 is the part of the positive circle 2 in

9、the first quadrant,and calculate curve integral 2LLxyxdyydx+=22202(2cos4sin)Lxdyydxtt dt=+22032(1sin)2t dt=+=2cosParametric equation of:,:022sinxtLtyt=Example 10Show that if is a positive integer,then 1lim0.pnn=2The equation of curve is 1|(1,1).The starting point is(1,0)and the ending point is(1,0).

10、Calculate LLyxxxydxx dy=+1Let:1,1,0,Lyx x=+the starting point is(1,0)and the ending point is(0,1);2:1,0,1,the starting point is(0,1)and the ending point is(1,0).Lyx x=012210=(1)(1)xxx dxxxx dx+12222LLLxydxx dyxydxx dyxydxx dy+=+1212=()()02323+=Exercises Lesson（III）for Triple integral,line and surface integral