二重积分的计算方法与技巧

二重积分的计算方法与技巧 [???????/]2006-07-17 [???6????]?????e(63),?z,?????x????,?????s???????????;?????????????1,?????%?????. ?????-??m] ???????-???????????;???? ?????e (?????s??????m??????;??????,????m?????m?530003) [????]???????-???????E?????????????g????,???E????m]?v???a?????????????-?????v ?????????????? ?3,?????????????(?????7??m[?????-???-?8???a?5???G?????8?????????l?a?????r????? f???????-?????? ??????????m] [???F??]???????-;???-?8??;?????r [?t???-????]G642.421m?[???F?7???[]Am? m?m????????-?????????????E?I???????????-?? ?????????-,?a????m[???-m[???????????(???????? ?am]?-?????????-??????,???????????????-???? ??,?E?????r???????v????????m]??1991???? 2005???%???/???????????????????2???-?2???? ???),???????-???2?????I???????????-??41%, ???????2????5.1%,?%???????-?????????????? ???2,???;???????-??????????????m]???E,???/?? ?????????-??????????????????,?5???????;?z?? ??m]???????????-??????,???F?a??????????????m] ???????????-??????????,???v???????????????f ?6???%?u??,?5?/????????;?f?6?.?(????,?????5 ?6m]???/???????;,???(?6?????????e??,???????? ???????)??m]?%?6?Z???I????????????,???????? ???????????-??????????m]?????????????-?????? ??????????????????,???v?????????-?????????? ???5??????m] m?m???m[???????????-?????????? ???????-???????(????????????????,???(?? ?????????-????????,?O???????9????m]???????? ???-?????v???????): ?v??,?Z???????????????-,???????-?8?????? ??; ?v??,?(???]???????7??; ?v?(,???????-?o???????????-?8?????5???G ??m]?????E?a???????7??,?O?????-?8???.???f?. ??????,?????????8???5???G??;?????E?a?????7 ??,?y?Z???????E?a?8?????fm[???3m[???????????? ?5???G??; ?v??,?-???????-?f?8?E?o???-???;????????; ?v??,?????????????7?????????;?8???????? ???????I???n????,???-?-?4?????g?G?8??,?a?3?? ?????????0?g?G?8???3?????????-,???-?????E?? ????m] m?m???m[?(???]???????7?? ???????????-??,?(???]???????7??,?O?????? ???????)????,???????????????5,?????(?????7?? ?5??,???????)?O?(???????M????,?I?????????? ??m]???7?????(??,?????????????????-?8???n?? ?????2??m]???v?????),???-?8???E?????f?(???? ?8??,?I?M?a???????7??????;???????????8f(x 2 + y 2 )m[f( x y )?ff( y x )?????G,?f?6???-?8???E?-??m[?? ??m[??????????????,?I?M?a?????7??????m] ?????????-I=?? D e y x+ydxdy,???tD?E?Ox=0,y =0??x+y=1?????4???8??m] ???8???-?e y x+ydx?;?e y x+ydy???5???a?(?????G ???8????,?????a???????7???)???-????????,???? ?Ge y x+y=e 1 1+x/y?m?8f( x y )?????G,???-?a?????7???) ?(??????????m]?????3,?-????x+y=1?f?8???? ?7???):2?= 1 cos2?+sin2? ,???-?8??:D:0s?2?s? 2? 2 ,0s?2? s? 1 cos2?+sin2? ?y I=? 2 0d2?? 1 cos2?+sin2? 0e sin2? cos2?+sin2?2?d2?= 1 2 ? 2 0e sin2? cos2?+sin2? 1 (cos2?+sin2?)2d2? 633 2006??10?1 ?v19?????3 ?????s?????????? JournalofGuangxiUniversityofFinanceandEconomics Oct.2006 Vol119m? ?? 19-: 1 2 ? 2 0e sin2? cos2?+sin2?d( sin2? cos2?+sin2? )= e-1 2 m?m??(m[?-???-?8???a?5???G?????8???? ?????t????,?5?.???/?5?????????-???-?8?? ?a?5???G?????8????,???E?c?4???????????-???? ???????????y???????r????m]????,?a???????7?? ?t,???-?8???5?f?????)?n?????v????: (1)???-?8???(?????f?n??????????????(???? 1) ????,???-?8??D?????mx???3,??x???8???8 [a,b],???????8????D?)m[?3???????)?-?U?8y= y1(x),y=y2(x)m]???8??[a,b]?????G????,?????? ??y????????,??D?????????n??,?y???n?????? ???7?-?U?8y1(x),y2(x),?????8?????8?4 D: as?xs?b y1(x)s?y2(x) ??1m?m?m?m?m?m?m?m???2 (2)???-?8???(?????f?n??????????????(???? 2) ???????-?8??D?????my???3,??y???8???8 [c,d],???????8????D??m[?????????)?-?U?8x= x1(y),x=x2(y)m]???8??[c,d]?????G????y,???? ????????????,?;D???????????n??,???n?????g ???7?-?U?8x1(y),x2(y),?y?8??D???8?4 D: cs?ys?d x1(y)s?xs?x2(y) ???U??,???-?8???E????????x=a,x=b,y= c,y=d?????4??????,?y?????8?4?5???G??:as?x s?b,cs?ys?d,?????G?(?????-?o??m]???????-?8?? ?E?O?????????[???????????4??,???((1)m[?????( (2)?0?-m] ????,?O????y=cx,x=b??x???????4?????? 2?(????3)?(?n??????y????,?????-???????.x ??,?9???5???G???82? s?xs? s?ys?x ?????O????y=x y= x 2 ,y=2?????4???(????D(????4),?(?????? ????x??,?-?(???????.y???;,?9??D: 0s?ys?2 ys?xs?2y ??3m?m?m?m?m?m?m???4 m?m???m[?l?a?????r?????f???? ???????-???z,?????-?8?????(???????????r ?%?????????8?f?????????f???????????????-,?l ?a?????r?????r?r???f????m] (1)???????-?8??D??????x??,D1?ED?ax ?????3?????-,?y ?? D f(x,y)d2?= 0m???f(x,y)??y?8??????, m???f(x,-y)=-f(x,y) 2?? D 1 f(x,y)d2?m???f(x,y)??y?8??????, m?m?m?m?m?m???f(x,-y)=-(x,y) (2)???????-?8??D??????y??,D2?ED?a??y ?????????-,?y ?? D f(x,y)d2?= 0m???f(x,y)??x?8??????, m???f(-x,y)=-f(x,y) 2?? D 1 f(x,y)d2?m???f(x,y)??x?8??????, m?m?m?m?m?m???f(-x,y)=f(x,y) (3)???????-?8??D??????????,D1m[D2???3???? ??,D1slD2???ED?a?v???????????-?8??,?y ?? D f(x,y)d2?= 0m???f(-x,-y)=-f(x,y) 2?? D 1 f(x,y)d2?=2?? D 1 f(x,y)d2? =4?? D 1 iAD 2 f(x,y)d2?m???f(-x,-y)=f(x,y) (4)???????-?8??D??????????y=x,??(x,y) s1DΖ(y,x)s1D(????5),D1,D2?-?U?E????y=x ???3m[?)?????-,?y ?? D f(x,y)d2?=?? D f(x,y)d2? ?? D 1 f(x,y)d2?=?? D 2 f(x,y)d2? 33 ???3?????e:???????-???????????;????10?1 ?? : ab 0c 7 (????5)m?m?m?m?m?m?m?m?(????6) ????,1991???%???/?????2???(??????:??D ?E?????3??A(1,1),B(-1,1),C(-1,-1)?8???? ???(????(????6),D1?E???a?v???????????-,?y?? D (xy+cosxsiny)dxdy=(?) (A)2?? D 1 cosxsinydxdy;m?m?m?m?(B)2?? D 1 xydxdy; (C)4?? D 1 (xy+cosxsiny)dxdy;m?m?m?m?(D)0m] ????6???8,????BO,??DcV1?8?(????AOB,D2 ?8?(????COB,????DcV1??????y??,D2??????y ??,?,xy??x??y???E??????,cosxsiny??x?E???? ??,??y?E??????,?? ?? D 1 xydxdy=?? D cV 1 xydxdy+?? D cV 1 xydxdy=0 ?? D cosxsinydxdy=?? D cV 1 cosxsinydxdy+?? D 2 cosxsinydxdy= 2?? D 1 cosxsinydxdy ???((A)m] ????,???t?-????4x 2 +y 2 =1?;????z=1-y ??z=0?????4????????????(????7)m] ?l?a???-?????????r?????????????????r,?v ?????????O???-????m]?O??????????????D?8?t ?-,????x????y????????,?%f(x,y)=y?Ey???? ????,???? V=?? D (1-y)d2?=?? D 1d2?+?? D yd2?=?fD(?t?-)?? ????= 2? 2 ?I?????3?????????)??,???????-?????????? ?????????E???8??,???????????-??????,???????? ??????????????????m] [???2???F] [1]??????,??????m???.????????(?v????)[M]. ?x??:??????????????,1964. [2]???4??,?????..?????-??(?)??)[M].?x??:?? ????????,2004. [3]?????cm???.????????:?5???7???;?????????- ??[Z],?8?????I??????,2002. [?i???f??:??m???] [?i???o??:?s????] 33 ?v19???????s??????????2006?? ?? 8