?= Ec×s x1×s￥à Q 1￡ ü?é 4?é 6 2￡ üμ?> u× ￥ ??f ?A V 3 !? ^ V ￥ ü ?m? bW ?8f;g?  ??￡ ü (1) ?? f(P) > 0, Of(P)·= 05R ? f(P)d? (2) ??￥ ???s u×?0 ‰ ? μ Z ? f(P)d? = Z ? g(P)d?; 5? μf(P) = g(P) 4 !f(x)[a,b] Vg(y)[c,d] V5f(x)g(y) ? u ×D[a;b]￡[c;d]  V O ZZ D f(x)g(y)dxdy = Z b a f(x)dx Z d c g(y)dy: 5 ?jf(x;y)jD  V * 1D  ^? V$ I3f ? f(x;y) = 8< : 1; ?x;y? ^μ ? ?; ?1; ?x;yà μB? ^í ? ? [0,1]￡[0,1] ￥s 6 !D = [0;1]￡[0;1] f(x;y) = 8 < : 1;x ^μ ? ? 0;x ^í ? ? ￡ üf(x;y)D ? V x2×s? ?Qs 1 1.9 ?/ =×s (1) RR D (y ?2x)dxdy;D = [3;5]￡[1;2] (2) RR D cos(x+y)dxdy;D = [0; …2]￡[0;…] (3) RR D xyex2+y2dxdy;D = [a;b]￡[c;d] (4) RR D x 1+xydxdy;D = [0;1]￡[0;1] 2.|=×sRR D f(x;y)dxdy?1?] ¨?￥ ?Qs (1) D?xàDx2 +y2 = r2(y > 0) ??? (2) D?y = x;x = 2#y = 1x(x > 0) ??? (3) D?y = x2;y = 2x3;y = 1?y = 2?? (4) D = f(x;y)jjxj+jyj6 1g 3.?M/  ?Qs￥Q? (1) R20 dyR3yy2 f(x;y)dx (2) R21 dxR2px f(x;y)dy (3) R10 dxRx20 f(x;y)dy +R31 dxR 1 2(3?x)0 f(x;y)dy. 4. !f(x;y) ?s￥ u×D  ??￡ ü Z b a dx Z x a f(x;y)dy = Z b a dy Z b y f(x;y)dx: 5.9 ?/ =×s (1) RR D xmykdxdy(m;k > 0);D ^?y2 = 2px(p > 0);x = p2??￥ u× 2 (2) RR D xdxdy;D ^?y = 0;y = sinx2;x = 0?x = p…??￥ u× (3) RR D pxdxdy;D : x2 +y2 6 x; (4) RR D jxyjdxdy;D : x2 +y2 6 a2; (5) RR D (x+y)dxdy;D?y = ex;y = 1;x = 0;x = 1 ??? (6) RR D x2y2dxdy;D?x = y2;x = 0;x = 2;y = 2+x ??? (7) RR D (x+y)dxdy;D ^[(2;2);(2;3)?(3;1)1??￥ ??? (8) RR D sinnxdxdy;D?y = x2;y = 4x?y = 4 ??? 6. p/ =×s (1) I = R10 dxR1x e?y2dy (2) I = R10 dxR1x x2e?y2dy (3) I = R p… 2 0 dy Rp… 2y y2 sinx2dx 7. !yà| ü ?μ? u×Ds??￥ ?sD1?D2￡ ü (1) ?f(x;y)1?x1 f ?'f(?x;y) = ?f(x;y)5 ZZ D f(x;y)dxdy = 0: (2) ?f(x;y)1?x1 }f ?'f(?x;y) = f(x;y)5 ZZ D f(x;y)dxdy = 2 ZZ D1 f(x;y)dxdy = 2 ZZ D2 f(x;y)dxdy: 8.9 ?/  ?×s (1) RRR V (x+y +z)dxdydz;V : x2 +y2 +z2 6 a2; 3 (2) RRR V zdxdydz;V? w ?z = x2 +y2;z = 1;z = 2 ??? (3) RRR V (1+x4)dxdydz;V? w ?x2 = z2 +y2;x = 2;x = 4 ??? (4) RRR V x3yzdxdydz;V ^? w ?x2 +y2 +z2 = 1;x = 0;y = 0;z = 0?? ￥ê??B?K￥μ? u× (5) RRR V xy2z3dxdydz;V? w ?z = xy;y = x;z = 0;x = 1 ??? (6) RRR V ycos(x + z)dxdydz;V ^?y = px;y = 0;z = 0#x + z = …2 ?? ?￥ u× 9.?M/  ?Qs￥Q? (1) R10 dxR1?x0 dyRx+y0 f(x;y;z)dz (2) R10 dxR10 dyRx2+y20 f(x;y;z)dz (3) R10 dxR10 dyR01?x?y f(x;y;z)dz (4) R1?1 dxR p1?x2 ?p1?x2 dy R1p x2+y2 f(x;y;z)dz 10 p/  ?8-8 (1) V?x2 +y2 +z2 6 r2;x2 +y2 +z2 6 2rz ? ?? (2) V?z 6 x2 +y2;y 6 x2;zV2 ? ?? (3) V ^?US ü ?#x = 2;y = 3;x+y +z = 4 ???￥??8 x3×s￥M }D 1.¨USMD|RR D f(x;y)dxdy?1 ?Qs (1) D??x2 +y2 6 a2;y > 0 4 (2) D?ìa2 6 x2 +y2 6 b2;x > 0 (3) D?x2 +y2 6 ay(a > 0) (4) D?Z?0 6 x 6 a;0 6 y 6 a 2.¨USMD9 ?/ =×s (1) RR D sinpx2 +y2dxdy;D : …2 6 x2 +y2 6 4…2 (2) RR D (x+y)dxdy;D ^?x2 +y2 6 x+y￥ =? (3) RR D (x2 +y2)dxdy;D? ? gL(x2 +y2)2 = a2(x2 ?y2)?? (4) RR D xdxdy;D?- ü￡ ?Lr = ?? L = …?? (5) RR D xydxdy;D? ? ?Lr = e ?? L = …?? 3./ s?? ??M u;v| ? ì?1 ?Qs (1) R20 dxR2?x1?x f(x;y)dy; ?u = x+y;v = x?y (2) Rba dxRflxfix f(x;y)dy(0 < a < b;0 < fi < fl); ?u = x;v = yx (3) RR D f(x;y)dxdy ?D = f(x;y)jpx + py 6 pa;x > 0;y > 0g ?x = ucos4 v;y = usin4 v (4) RR D f(x;y)dxdy ?D = f(x;y)jx + y 6 a;x > 0;y > 0g(a > 0) ?x+y = u;y = uv 4.T a?￥M }D p/ s (1) RR D (x2 +y2)dxdy;D ^?x4 +y4 = 1??￥ u× (2) RR D (x+y)dxdy;D?y = 4x2;y = 9x2;x = 4y2;x = 9y2?? 5 (3) RR D (x2 +y2)dxdy;D?xy = 2;xy = 4;y = x;y = 2x?? 5. ?¨=×s p?/  w ???￥ ?8￥8 (1) z = xy;x2 +y2 = a2;z = 0 (2) z = hRpx2 +y2;z = 0;x2 +y2 = R2 (3) o ?x2 +y2 +z2 = a2D?? ?x2 +y2 = ax(a > 0)￥ ?s (4) x2a2 + y2b2 + z2c2 = 1; x2a2 + y2b2 = z2c2(z > 0) (5) z2 = x24 + y29 ;2z = x24 + y29 (6) z = x2 +y2;z = x+y 6. p wL(x2a2 + y2b2)2 = xyc2 ???￥ ? 7.¨?USMD9 ?/  ?×s (1) RRR V (x2 +y2)dxdydz;V? w ?z = x2 +y2;z = 4;z = 16?? (2) RRR V (px2 +y2)3dxdydz;V? w ?x2 + y2 = 9;x2 + y2 = 16;z2 = x2 + y2;z > 0?? 8.¨ oUSMD9 ?/  ?×s (1) RRR V (x+y +z)dxdydz;V : x2 +y2 +z2 6 R2 (2) RRR V (px2 +y2 +z2)5dxdydz;V?x2 +y2 +z2 = 2z?? (3) RRR V x2dxdydz;V?x2 +y2 = z2;x2 +y2 +z2 = 8?? 9.T a?￥M }D p/  ?×s (1) RRR V x2y2zdxdydz;V?z = x2+y2a ;z = x2+y2b ;xy = c;xy = d;y = fix;y = 6 flx??￥ ?8 ?0 < a < b;0 < c < d;0 < fi < beta (2) RRR V x2yzdxdydz;V](1) (3) RRR V y4dxdydz;V?x = az2;x = bz2;(z > 0;0 < a < b);x = fiy;x = fly(0 < fi < fl)[#x = h(> 0)?? (4) RRR V e q x2 a2+ y2 b2 + z2 c2 dxdydz;V?x 2 a2 + y2 b2 + z2 c2 = 1?? (5) R10 dxR p1?x2 0 dy R2?x2?y2p x2+y2 z 2dz: 10 p/ ò w ? ?? ?8-8 (1) z = x2 +y2;z = 2(x2 +y2);y = x;y = x2 (2) (xa + yb)2 +(zc)2 = 1(x > 0;y > 0;z > 0;a > 0;b > 0;c > 0): x4 w ? ? 1. p/  w ?￥ ? (1) z = axyc??x2 +y2 = a2 =￥?s (2) ?x2 +y2 = 13z2D ü ?x+y +z = 2a(a > 0) ???s￥V ? (3) ?z = px2 +y2$? ?z2 = 2x ???s (4) w ?z = p2xy$ ü ?x+y = 1;x = 1#y = 1 ??/￥?s 2. ?è ?x = rcos’;y = rsin’;z = h’(0 < r < a;0 < ’ < 2…)￥ ? 3. pì ?x = (b + acos?)cos’;y = (b + acos?)sin’;z = asin?(0 < a 6 b)$ HüL’ = ’1;’ = ’2? H?L? = ?1;? = ?2 ????s￥ ? i p??ì ?￥ ? x5×s￥t ??¨ 7 1. p/  ( á￥ ü ? e￥é? (1)???x2a2 + y2b2 6 1;y > 0 (2)ú1h?sY1a?b￥?%0? (3) r = a(1+cos’)(0 6 ’ 6 …) ??￥ e (4) ay = x2;x+y = 2a(a > 0) ??￥ e 2. p/  á ( ￥t8￥é? (1) z 6 1?x2 ?y2:z > 0 (2)?US ?# ü ?x+2y ?z = 1 ???￥ 1 ?8 (3) z = x2 +y2;x+y = a;x = 0;y = 0;z = 0??￥ ?8 (4) z2 = x2 +y2(z > 0)? ü ?z = h??￥ ?8 (5)? o Ta2 6 x2 +y2 +z2 6 b2;z > 0. 3. p/  á ( ￥ ü ? e￥?8  (1)Hé1a?b OC?1’￥ ü? 1H?1??Hb￥?8  (2) y = x2;y = 1 ?? ü ?m?1?°Ly = ?1￥?8  4. p?/  w ? ?? ( 8￥?8  (1) z = x2 +y2;x+y = §1;x?y = §1;z = 01?zà￥?8  (2)éZ81? ?￥B í￥?8  (3)?ca2 6 x2 +y2 6 b2;?h 6 z 6 h1?xà?zà￥?8  8 5. ! o8x2 + y2 + z2 6 2x ò?￥ á?????USe?￥  ? p? o￥é  6. p (  ?x2 +y2 6 R2;z = 0xà B?(0;0;c)(c > 0))?êé ?￥? ? 7. p ( ?8x2 + y2 6 a2;0 6 z 6 h?(0;0;c)(c > h))?êé?￥ ? ? 9