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!=2xy dxx2y 2dy ???U=f(x;y)2R2; y >0g? C [AB]??A(1;2)????B(3;4)? !(x;y) =2xy(1 +x2)2dx+(x)dy? ???? ???????Z C != (x2+y2a2)dx2aydy; ??a??? ?? ?????? ???? ??? ???? C != (3x2y+z3)dx+ (3y2z+x3)dy+ (3xz2+y3)dz? x2+y2+z2= 1 x+y+z= 0? y2dx+x2dy??????? 2+y2b 22xa
2yb C !=xdyydxx 2+y2? C ??!=ysinxdx+xcosydy??C??? ?? ??????? ?? ??????(OA)??O(0;0)????A(1;1)?
+xyy22 ? ????Z C f(0;0)f(2;0) =2? :@f@x (x;y) =2xy(1 +x2)2 @f@y (x;y) =11 +x2? ?? ??????H0(x) = 0? ???? f(x;y) =y1 +x2 ???? ??????P(x;y) =x2+y2a2??Q(x;y) =2ay? ?? ?@P@y (x;y) = 2y??@Q@x (x;y) = 0? ???? ???? ??? ?? ????? ???? ??????? ?? ??????P1(x;y) =f(x)P(x;y)??Q1(x;y) =f(x)Q(x;y)? ?? @P 1@y (x;y) =f(x)2y=@Q1@x (x;y) =f0(x)Q(x;y) =f0(x)2ay?
t4+23 t3 1 0 10130
Z [OP]!F :!dM=Z 1 0 (tt)12t2t2dt=Z 1 0
(x;y) = 2x3??@P@y (x;y) =y22
a4bab23 ???? ???? ??K??? ??????? ?? ????? ??????? ?C1=f(t;0); t2[0;1]g?C2=f(cost;sint); t2[0;2 ]g ??C3=f(0;t); t2[1;0]g? I=Z !=Z 2
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!=2xy dxx2y 2dy ???U=f(x;y)2R2; y >0g? C [AB]??A(1;2)????B(3;4)? !(x;y) =2xy(1 +x2)2dx+(x)dy? ???? ???????Z C != (x2+y2a2)dx2aydy; ??a??? ?? ?????? ???? ??? ???? C != (3x2y+z3)dx+ (3y2z+x3)dy+ (3xz2+y3)dz? x2+y2+z2= 1 x+y+z= 0? y2dx+x2dy??????? 2+y2b 22xa
2yb C !=xdyydxx 2+y2? C ??!=ysinxdx+xcosydy??C??? ?? ??????? ?? ??????(OA)??O(0;0)????A(1;1)?
2+y2dx+x+yx
2+y2dy?? ???? ?? ?????ABCD? ????A(1;1)?
???? ??????? ?? ??????[OB]??O(0;0)????B(1;1)? ??????? ??0?2? (yz)dx+ (zx)dy+ (xy)dz x2+y22x0 x2+y22y0
zdx+xdy+ydz?? ????V(x;y) =yx2+y2;xx
2+y2 x 2!k? ?? : (x=t2;y= 2t;z=t)? ???? ???????[OP]? ??!F=xx2+y2+ 1;yx
2+y2+ 1
??!F= (2xy2z;2x2yz;x2y22z)??C??? ?? ?????? ?????? ???x= cost?y=p3 2 sint? ??z=12 sint ????0t2? Z (2xyx2)dx+ (x+y2)dy ????D=f(x;y)2R2; x0; y0x2a 2+y2b J=Z Z D (2x3y)dxdy? ????K=f(x;y)2R2; x0; y0; x2+y21g? ???? y=asin3t????t2[0;2]? ????t2[0;2]? x=a(tsint) y=a(1cost)????0t2? 8>< :x=acost y=asint z=ht ????0t2 ??????P(x;y) =yx2+y2??Q(x;y) =xx
2+y2? ?? ?????? ???@P@y
=@Q@x =y2x2(x2+y2)2? @f@x (x;y) =yx2+y2??@f@y
(x;y) =xx 2+y2? ) +H(x)??H???C1?? ?? ?????? ??? ??x? ?? ? ?????@f@x
(x;y) =H0(x)yx2+y2=yx
2+y2? ????? ?f(x;y) =??????(yx ???? ? ??????@P@y (x;y) =@Q@x (x;y) =2xy @f@x (x;y) =2xy ??@f@y (x;y) =x2y +H(y)?? ?? ??????? ???? ?????H0(y) = 0? ???? Z C !=f(3;8)f(1;2) =58 ?? ? ??????@P@y (x;y) =@Q@x (x;y) = 3y212xy? ???? ?? ????? ??? ?????? ???R2??? ??? ??????? ???? Z C !=f(B)f(A) =236? (x;y) =@Q@x (x;y) = 1? x 22+xyy22 ? ????Z C f(0;0)f(2;0) =2? :@f@x (x;y) =2xy(1 +x2)2 @f@y (x;y) =11 +x2? ?? ??????H0(x) = 0? ???? f(x;y) =y1 +x2 ???? ??????P(x;y) =x2+y2a2??Q(x;y) =2ay? ?? ?@P@y (x;y) = 2y??@Q@x (x;y) = 0? ???? ???? ??? ?? ????? ???? ??????? ?? ??????P1(x;y) =f(x)P(x;y)??Q1(x;y) =f(x)Q(x;y)? ?? @P 1@y (x;y) =f(x)2y=@Q1@x (x;y) =f0(x)Q(x;y) =f0(x)2ay?
2ayf0(x) =f(x)2y?
?? ?? ?????? ???? ???f0(x) =f(x)a? ?? ??? ?????f(x) = exp(xa 8>< :@F@x (x;y) = exp(xa )(x2+y2a2) @F@y (x;y) =2ayexp(xa exp( xa )y2+H0(x) = exp(xa )(x2+y2a2))H0(x) =x2exp(xa )a2exp(xa ?? ????P(x;y;z) = 3x2y+z3?Q(x;y;z) = 3y2z+x3??R(x;y;z) = 3xz2+y3? ?? ?????? ??? ?? @P@y =@Q@x ;@P@z =@R@x ??@Q@z =@R@y (y;z) = 3y2z+x3? ?? ?? ?????? ???g(y;z) =y3z+h(z)??h f(x;y;z) =x3y+xz3+y3z+c; c2R yz+zx? ????!??? ?????? ?? ?????C??? ??????? ???? ?????0? )2= (a2 ??????(0;a2 )?? ?? ?????a2 cos??y=a2 +a2 sin? ?? ? ????? ? Z y2dx+x2dy=Z 2 0 a2 +a2 sin 2a2 sin +a24 cos2a2 cos d Z 20a2sin22
d a24 Z 201cos22
d a34 ?(xa)2a2+(yb)2b
???x=a(1 +p2cos)??y=b(1 +p2cos)? ?? ? ?????? Z y2dx+x2dy=Z 2 0 b2(1 +p2sin)2(ap2sin) +a2(1 +p2cos)2(bp2cos)d Z 2 04ab2sin2+ 4a2bcos2d
=4ab2+ 4a2b = 4ab(ab): ?? ?Z C !=Z C 1!+Z C 2!+Z C 3!+Z C 4!? t2[a;a]? ?? ? ???? ? Z C 1!=Z a ax(t)y0(t)y(t)x0(t)x2(t) +y2(t)dt
Z a a0 +aa2+t2dt=
??????ta a a =??????(1)??????(1) =2 Z C != 2? ??Z C !=Z 11(x3+ (x+x2)2x)dx=694
Z C !=Z 1 0 x(sinx+ cosx)dxIPP=[x(cosx+ sinx)]10Z 1 0 (cosx+ sinx)dx= 2sin11? Z C !=Z [AB]!+Z [BC]!+Z [CD]!+Z [DA]! Z 11x1x+ 1dx+Z
111 +y1 +y2dy+Z
11x+ 1x
2+ 1dx+Z
111 +y1 +y2dy
= 4 Z 11dx1 +x2= 2:
??Z C 1!=Z 1 0 (x2x2)dx= 0? Z C 2!=Z 1 0 (2y5y3)dy=112 I=Z 2 0 (y(t)z(t))x0(t) + (z(t)x(t))y0(t) + (x(t)y(t))z0(t)dt Z 2 0 (R2+hRt(cost+ sint) +hR(costsint))dt =2R(h+R): C 1:( x= cost y= 1 + sint????t2[2 ;0]C2:( x= 1 + cosu y= sinu????u2[2 Z C !=Z C 1!+Z C 2! Z 0 2 ((1 + sint)sint+ 2cos2t)dt+Z 2 (sin2u+ 2(1 + cosu)cosu)du 2 1: ?? ????P(x;y) =x+y??Q(x;y) =xy? ?? ??? ????? ???@P@y (x;y) =@Q@x (x;y) = 1? @f@x (x;y) =x+y)f(x;y) =x22 +xy+H(y)?? @f@y (x;y) =x+H0(y) =xy)f(x;y) =x22 +xyy22 C !=f(0;0)f(2;0) =2: 12 8>< :x=12 +12 cos y=1p2 12 12 cos ????2[0;2]? ???? Z zdx+xdy+ydz=Z 2 0 (12 12 cos)(12 sin) + (12 +12 cos)(1p2 cos+1p2 sin12 sind Z 2 012 p2 (cos2+sin2)d=p2 2V(x(t);y(t)) = (sin(t)R
;cos(t)R (x0(t);y0(t)) = (Rsint;Rcost)? ZC!V :!dM=Z
sin2t+ cos2tdt= 2: Z !F :!dM=Z 1 0 (t2t)2t3t22t2 +t4(1)dt Z 1 0 t410t32t2dt =t55 +52t4+23 t3 1 0 10130
Z [OP]!F :!dM=Z 1 0 (tt)12t2t2dt=Z 1 0
13t2dt=133
ZC!F :!dM=Z
0 (abcos2t+absin2t)dt=ab? =f(x;y)2R2;0x1; x2ypxg? ?? ????P(x;y) = 2xyx2??Q(x;y) =x+y2? ?? ? ????? ? I=Z Z @Q@x @P@y dxdy Z 1 0Z px x2(12x)dydx
Z 1 0 (12x)(pxx2)dx 130(x;y) = 2x3??@P@y (x;y) =y22
P(x;y) =y22
I=Z @DPdx+Qdy
Z 2 0b2sin22
(asin) +b4cos42 (bcos)d 415a4bab23 ???? ???? ??K??? ??????? ?? ????? ??????? ?C1=f(t;0); t2[0;1]g?C2=f(cost;sint); t2[0;2 ]g ??C3=f(0;t); t2[1;0]g? I=Z !=Z 2