高数二重积分习题加答案

0yx112x+y=12x+y1由重积分的性质I1I2二重积分习题课例1.2)1()2(,)()(223221围成由其中的大小与比较yxDdyxIdyxIDD0yxD:x+y=1,x–y=1，x=0所围.11–1先对y积分xxyyxfId),(y=1–xy=x–1xd例2将二重积分化成二次积分,d)d,(DyxyxfI0yx11–1先对x积分21DDID1D2xyxfyyd),(dxyxfyyd),(dx=1–yx=y+1ID:22xaxyx.d),(d022axaxxyyxfxI0yxaax0ayda222xaxyaaxy)(即,ax又22yaax22yaaxyyaaxyxf)d,(例3将二次积分换序.)d,(d20202变为极坐标形式将RyRyxyxfyI2R区域边界：x=0I0yx即r=2Rsinr=2Rsin20sin20d)sin,cos(dπθRrrθrθrfθ例422yRyx2即πθD例5解围成．由其中计算2,1,,22xxyxyDdyxDxxDdyyxdxdyx12221222112dxyxxx213)(dxxx.49.21,1:xxyxD例6解.10,11:,2yxDdxyD其中计算1D2D3D先去掉绝对值符号，如图dxydyxdxyDDDD321)()(2221211021122)()(xxdyxydxdyyxdx.1511.d)()(11d)()(d12banxanbayyfybnyyfyxx证xanbayyfyxxd)()(d2.d)()(111banyyfybn例7证明Dxyoxybaabxyfyxynd)()(d2abybbabyndyyxnyf1)(11)(例8计算解10sin)1(xdxx.1sin1yydxxxdysin10xxdyxxdx2sin10.sin10yydxxxdy.dd,2222222yxbyaxRyxDD求为圆域设yxbyaxDdd2222.1141224baR解R由对称性xOyyxyyxxDDdddd22yxyxDdd)(2122Rrdrrba022022d1121yxyxbaDdd)(11212222例9yxzo例10.)0(2322222所围成立体的表面积与旋转抛物面求半球面aazyxyxazxyzoDS=1S2S共同的D:azyxyxaz2322222a2zayx即2S2S2S1S1S2221:3Szaxy222:2xySza2211zzdAdxdyxy22233adxdyaxy所求面积：2221zzdAdxdyxy222axydxdya2221222233DDaxyaAAAdxdydxdyaaxy2222222200001133aaadrdrdarrdraar222222001263aaardrarrdraar22222022222013(3)3()aaadararardara2422636.33a练习题计算下列二次积分：交换下列二次积分的次序:;),(),(.130312010yydxyxfdydxyxfdy;),(.221110xxdyyxfdx;.322222202020yRxRRyyxRydxedyedxedye.ln1.4551yxdxdyy23021.(,)xxdxfxydy2212200102.(,)(,)yyydyfxydxdyfxydx练习题答案222043.(1).8RrRIderdre5511114.ln4.lnlnxdyIdxxdxyxx11,]1,0[)(上的正值连续函数为设x)(21dd)()()()(bayxyxybxaD证明：为常数，其中ba,证yxyxybxaIDdd)()()()(设的对称性得由区域关于直线xyyxxyxbyaIDdd)()()()(所以,DyxbaIdd)(2).(21baIxybaxyO则对称关于直线若闭区域,xyDDDxyfyxfdd),(),(}.,),{(10yxyxD