同济大学高等数学课件D92二重积分的计算.ppt

*三、二重积分的换元法第二节一、利用直角坐标计算二重积分二、利用极坐标计算二重积分机动目录上页下页返回结束二重积分的计算法第九章一、利用直角坐标计算二重积分且在D上连续时,0),(?yxf当被积函数???????bxaxyxD)()(:21????Dyxyxfdd),(yyxfxxd),()()(21?????baxd由曲顶柱体体积的计算可知,若D为X–型区域则)(1xy??)(2xy??xboyDax若D为Y–型区域???????dycyxyD)()(:21??y)(1yx??)(2yx??xdocyxyxfyyd),()()(21????dcyd则机动目录上页下页返回结束当被积函数),(yxf???2),(),(),(yxfyxfyxf2),(),(yxfyxf?),(1yxf),(2yxf均非负在D上变号时,:(1)若积分区域既是X–型区域又是Y–型区域,??Dyxyxfdd),(为计算方便,可选择积分序,必要时还可以交换积分序.)(2xy??xoyDba)(1yx??)(2yx??dc则有x)(1xy??yyyxfxxd),()()(21?????baxdxyxfyyd),()()(21?????dcyd(2)若积分域较复杂,可将它分成若干1D2D3DX-型域或Y-型域,???????????321DDDD则机动目录上页下页返回结束xy211xy?o2??,d???DyxI?其中D是直线y=1,x=2,及y=–型区域,则???:D?I?21dx?yyxd??21dx?????2121321dxxx89???–型区域,则???:D?I?xyxd?21dy??yyx2221?????21321d2yyy89?y1xy2xy??121??x2??xy21??,d??Dyx?:为计算简便,先对x后对y积分,???:D?xyxd???Dyx?d???21dy??????212221d2yyxyy?????2152d])2([21yyyyDxy?22??xy21?4oyxy22???yxy21???y2y2?,ddsin?????xxy?解:由被积函数可知,因此取D为X–型域:????????xxyD00:???Dyxxxddsin?xy0d???0dsinxx2????0dsinxxx先对x积分不行,说明:有些二次积分为了积分方便,???????22802222020d),(dd),(dxxyyxfxyyxfxI解:积分域由两部分组成:,200:2211???????xxyD822??yx2D22yxo21D221xy?2????????22280:22xxyD21DDD??将???:D视为Y–型区域,则282yxy???20??y???DyxyxfIdd),(??282d),(yyxyxf??,42xy??1,3?????xy3??2D1D1?x解:令)1ln(),(2yyxyxf???21DDD??(如图所示)显然,,1上在D),(),(yxfyxf???,2上在D),(),(yxfyxf???yxyyxIDdd)1ln(12??????0?yxyyxDdd)1ln(22?????4机动目录上页下页返回结束xyokkkrr?????kkkkkkrr????sin,cos??对应有二、利用极坐标计算二重积分在极坐标系下,用同心圆r=常数则除包含边界点的小区域外,小区域的面积k??),,2,1(nkk????在k??),,(kkr?k???kk??????krr?k??kkr????221内取点kkkrr??????221)(及射线?=常数,分划区域D为krkr?k??kkr??机动目录上页下页返回结束