?= EBc wLsD w ?s x1?B? wLsD w ?s 1v×s￥'?é?B? wLs??B? w ?s￥ ? ??é 29 ?/ ?B? wLs (1) RL (x2 +y2)ds ?L ^[ 00 20 011?? ￥ ??? (2) RLpx2 +y2ds ?L ^??x2 +y2 = ax (3) RL xyzds ?L1 ?Lx = acost;y = asint;z = bt(0 < a < b);0 ? t ? 2… (4) RL (x2 +y2 +z2)ds ?LD(3)M] (5) RL (x43 +y43)ds ?L1 =?Lx23 +y23 = a23 (6) RL y2ds ?L1?L￥B x = a(t?sint);y = a(1?cost);0 ? t ? 2… (7) RL xyds ?L1 o ?x2 +y2 +z2 = a2D ü ?x+y+z = 0￥?L (8) RL (xy +yz +zx)ds ?L](7) (9) RL xyzds ?L ^ wLx = t;y = 23p2t3;z = 12t2(0 ? t ? 1) (10) RLp2y2 +z2ds ?L ^x2 +y2 +z2 = a2Dx = yM?￥?? 39 ?/ ?B? w ?s (1) RR S (x2 +y2)dS ?S ^ ?8px2 +y2 ? z ? 1￥H? w ? 1 (2) RR S dS x2+y2 ?S1? ?x 2 + y2 = R2$ ü ?z = 0?z = H ?? |￥ ?s (3) RR S jx3y2zjdS ?S1 w ?z = x2 +y2$z = 1é/￥?s (4) RR S z2dS ?S1 ?è ?￥B?s x = ucosv;y = usinv;z = v (0 ? u ? a;0 ? v ? 2…) (5) RR S (x2 +y2)dSS ^ o ?x2 +y2 +z2 = R2 4 ! wLL￥Z?1 x = et cost;y = et sint;z = et (0 ? t ? t0) ? ?B?￥ áD??￥ O? üZ?Q1 O?(1,0,1))11 p ? ￥é  5 !μBé s?? ( ￥???x = rcos ;y = rsin (0 ? ? …) L በ= a (a1è ?) p ?e?(00))é 1m￥é?￥? ? 6 p ?L￥B|Lx = acost;y = asint;z = h2…t(0 ? t ? 2…)xà￥ ?8 I = RL (y2 +z2)ds !N ?L￥L á ^ ( ￥ 7 p ?t ? Tz = 12(x2+y2)0 ? z ? 1￥é  !N T￥ በ= z 89 ? o ? ???x2 + y2 + z2 = a2x > 0;y > 0;z > 0￥?L￥×? US !L በ= 1 9 p ( o Tx2 +y2 +z2 = a2(z ? 0)zà￥?8  10 p ( o ?z = pa2 ?x2 ?y2(x ? 0;y ? 0;x+y ? a)￥×?US 11 ? wL[USó‰ = ‰( )( 1 ? ? 2) kó9 ?RL f(x;y)ds￥ Ti¨N T9 ?/  wLs 2 (1) RL e p x2+y2ds ?L ^ wL‰ = a(0 ? ? … 4) (2) RL xds ?L ^ ? ?L‰ = aek (k > 0)?r = a =￥?s 12 p በ= ‰0￥?? ?x = rcos’;y = rsin’;z = r(0 ? ’ ? 2…;0 < b ? r ? a)ê? w ???(0,0,0)￥?êé?￥? ??b ! 0 H 2T ??$ 139 ?F(t) = RR S f(x;y;z)dS ?S ^B ü ?x+y +z = t7 f(x;y;z) = 8 < : 1?x2 ?y2 ?z2; 2 +y2 +z2 ? 1; 0; 2 +y2 +z2 > 1: : x2?=? wLsD w ?s 19 ?/ ?=? wLs (1) RL (2a?y)dx+dy ?L1?Lx = a(t?sint);y = a(1?cost);(0 ? t ? 2…)t9F￥Z_ (2) RL ?xdx+ydyx2+y2 ds ?L1??x2 +y2 = a2G I H?Z_ (3) RL xdx+ydy +zdz ?L1V 1,1,1? 2,3,4￥°L  (4) RL (x2 ?2xy)dx+(y2 ?2xy)dyL1y = x2V(1,1)?(-1,1) (5) RL ydx?xdy +(x2 +y2)dzL1 wLx = et;y = e?t;z = atV(1,1,0)?(e;e?1;a) (6) RL (x2 +y2)dx+(x2 ?y2)dyL1[A(1;0);B(2;0);C(2;1);D(1;1)1 ??￥?Z? I H?Z_ 29 ? wLs Z L (y2 ?z2)dx+(z2 ?x2)dy +(x2 ?y2)dz 3 (1) L1 o ? ???x2 +y2 +z2 = 1x ? 0;y ? 0;z ? 0￥H?LV o ￥?? A ?L￥Z_1 I H?Z_ (2) L ^ o ?x2+y2+z2 = a2?? ?x2+y2 = ax(a > 0)￥?Lê?Oxy ü ? Z￥?sVxà (b;0;0)(b > a)? A ?L ^ ¨ H?Z_ 3 p> wLL ￥?=? wLs I L ydx?xdy x2 +y2 (1) L1?x2 +y2 = a2 I H?Z_ (2) L1??x2a2 + y2b2 = 1 ¨ H?Z_ (3) L1[(0,0)1??Hé1aH ü??USà￥?Z? ¨ H ?Z_ (4) L ^[(?1;?1);(1;?1);(0;1)1??￥ ??? ¨ H?Z_ 4 p ??F?￥?êé? ?T￥?Né? wLLVA?? ?B? (1) F = (x?2xy2;y ?2x2y)L1 ü ? wLy = x2A(0;0);B(1;1) (2) F = (x+y;xy)L1 ü ? wLy = 1?j1?xjA(0;0);B(2;0) (3) F = (x ? y;y ? z;z ? x)L￥ O ? T1r(t) = ti + t2j + t3kA(0;0;0);B(1;1;1) (4) F = (y2;z2;x2)L￥? ? T1x = ficost;y = flsint;z = t fi;fl; 1? ?A(fi;0;0);B(fi;0;2… ) 5 !P;Q;RL  ??L1;á? ?é1l￡ ü j Z L Pdx+Qdy +Rdzj? Ml 4 ?M = max (x;y;z)2L np P2 +Q2 +R2 o  6 !;á> wLL;á w ?S S￥Z?1z = f(x;y) w LLOxy ü ? ￥g? wL1lf ?P(x;y;z)L  ??￡ ü I L P(x;y;z)dx = I l P(x;y;f(x;y))dx 79 ?I = RL xyzdz ?Lx2 +y2 +z2 = 1Dy = zM?￥? Z _? wLGQüV1,2,7,8?K 89 ?/ ?=? w ?s (1) RR S y(x?z)dydz +x2dzdx+(y2 +xz)dxdy ?S1x = y = z = 0x = y = z = a B? ü ? ??￥? ?Z8￥?? (2) RR S (x+y)dydz +(y +z)dzdx+(z +x)dxdy ?S ^[e?1? ?Hé12￥? ?Z8V ?￥?? (3) RR S yzdzdxS1x2a2 + y2b2 + z2c2 = 1￥ ??s￥ ? (4)RR S zdxdy +xdydz +ydzdxS1? ?x2+y2 = 1$ ü ?z = 0#z = 3 ? ??s￥?? (5) RR S xydydz +yzdzdx+xzdxdyS ^? ü ?x = y = z = 0?x+y+z = 1 ??￥ 1 ?8V ?￥?? (6) RR S x3dydz +y3dzdx+z3dxdyS1 o ?x2 +y2 +z2 = a2￥?? (7) RR S x2dydz +y2dzdx+z2dxdyS ^ o ?(x?a)2 +(y?b)2 +(z?c)2 = R2￥?? 9 !  @8￥ @ ?1v = (k;y;0) p?ê HW =V o ?x2 +y2 +z2 = 4￥ =? @V o ?￥ @  5 10 ! @8￥ @ ?1v = (xy5;0;z5xx) p,V? ?x2 + y2 = a2(?h ? z ? h)??￥ @  6