強(qiáng)化七天高數(shù)專題

21bfx，ydxdy21bfx，ydxdyfx，yaDydcDydxdy y，， 1fx，ydxdyfx，ydxdyfrcos，rsinDDfrcos，rsin2d1(（1）xydxdy,D:x（1）xydxdy,D:xy1,x0,yD1dx1x00112x(1x)(1x)203（2）ydxdy,D:x2y22x,yD20213rsin（2）ydxdy,D:x2y22x,yD20213rsind208322cos3sinsin3==rd00208323232cos3dcos4=0xy（3D{(x,y)|1x2y2y3arctanxD3解xy（3D{(x,y)|1x2y2y3arctanxD3解3arctantan16363633rdrd11362r31 22是由rsin2(0)2D2是由rsin2(0)2D2sin解drcosrsin002sin0r121842sin2502t18t25sin2sin58200211852t18t25sin2sin5820021185001421 （5D是由y軸圍1x2,x2(y1)21,右（5D是由y軸圍1x2,x2(y1)21,右上方區(qū)域，求：D解x22yr2(y1)21x2rsinr226rcosrsin269max{xy,}xy,xymax{xy,}xy,xyD例2ax{xy,dxdyxydxdyDDxy1dxdyDxy1dxdyxy1dxdyD221dx1xy1dyx219ln4x2y21,x2y2x2y21,x2y2D例2.|y21|11(xy2),x2y22D2y21||Dy21dxdy1(x2y2x2y2Dy221(x2x2D1dxdy21(x2x2D1dxdy21(x2y22111(1ry2220000134xy,11x2y2xy,11x2y2y2]例3.xy[12xy,21x2y2222xy,1x2y2y2]dxdyxydxdyxy[1y2]dxdyxydxdyxy[1D21drcosrsin0102d2rcosrsin018練習(xí)1（03三）a練習(xí)1（03三）a0a,0xf(x)g(x)0其D表示全平面，求If(x)gyD解a2,0x1,0yxf(x)g(yx)0,其If(x)g(If(x)g(yD12a0xf(x,y) ,1|x|f(x,y) ,1|x||y|f(x,DD:|x||y|20解 11cosx2dy41r0cos142 2101112dxydye例dxe00x01122y101112dxydye例dxe00x01122yy2ed01e21(1e1)1041例df(rcos,41例df(rcos,rsin0012dyf(x,0x2y2練習(xí)（12224022402f(x,f(x,0y0111yf(x,y)dx2dy2f111yf(x,y)dx2dy2f(x,y)dx（3）014yy1x2dxx2f(x,0例3（07一）變換積分1例3（07一）變換積分1sinf(x,2arcsin1dyf(x,0例4（10二）計(jì)算rD1r2cos2D:0例4（10二）計(jì)算rD1r2cos2D:0rsec,04r1(r2cos2r2sin2D1dx1x2y21y200D11dx1x2y2d(1x2y2 01x0121x2y2d(1x2y2dx013y2131131x0121x2y2d(1x2y2dx013y2131133(11(1x2)2x01313(12cos3dsin(12cos4d0013題型三例1.（1I1（2I22題型三例1.（1I1（2I22311ydy21010230022x22.（1I（2I例3.（1例3.（1I1（3I（2I22x2I例4.（1）（例4.（1）（2）x2ydxdyDDy2dxdy2D（3）D綜合練習(xí)：（1）I綜合練習(xí)：（1）I(xy)2Dx22xyD4x2y2dxdy8xy綜合練習(xí)xyxy綜合練習(xí)xy4x綜合練習(xí)：（3）(綜合練習(xí)：（3）(xycosxsiny)dxdyDD3(xycosxsiny)dxdy(xycosxsin2cosxsinydxdy答案綜合練綜合練習(xí)答案綜合練習(xí)：（5）1)dxdy綜合練習(xí)：（5）1)dxdyD(xy51)dxdy(xy5D3D221dxdy21dxdy答案綜合練習(xí)f(x)f(y)If(y)df(x)f綜合練習(xí)f(x)f(y)If(y)df(x)f(x)f(y)f(fDDf(x)f(y))f(x)2If(y))f(x)f(D答案abd(aD綜合練習(xí)y)dxdyx2dxdy綜合練習(xí)y)dxdyx2dxdy4I(x2DD21d2x2y20021（D第一象限 41綜合練習(xí)：（8）（01三1(x2y2綜合練習(xí)：（8）（01三1(x2y2y[1DD是由yxy1及x11(x2y2解：dxdyD1(x2y22y[1]dxdyydxdyDD題型四ux例D:0x題型四ux例D:0x1,0yvy:2u3,1v0u21,0v1uvdv41130(x2)(y002例2.解法一42Dy2dy00例2.解法一42Dy2dy003 (x2y 403043 (x2y 03041x221x2243402112 32x40143 (x2y 03041x221x2243402112 32x401x2dx1x243222x34400x2x2433221x244002243x2sin4sin2tcostd2sincostd2sin3x2x2433221x244002243x2sin4sin2tcostd2sincostd2sin3002283tdt222sintcos3008332cos2tdt2(1cos2t)cos20028332cos2tdt(1cos2t)cos200332cos2tdt2cos40028332cos2tdt(1cos2t)cos200332cos2tdt2cos4001 313 42xy2Du解法二42:u2xy2Du解法二42:u2Dy2dxdy28u2dudv210u2dudv5u2521rrdr2d200D{(x,y)|(x1)2(y1)22,yux例vD{(x,y)|(x1)2(y1)22,yux例vyD{(u,v)|v2,v/222 d(rcosrsin(xu4D2cossind r2dr404442283(sincos32yy222yy22(y1)21(x例4.xu(y1(x0)u2v21(uvyIv1dudvdudvS(D/)42//DD題型五f(題型五f(x,y)g(x,f(x,y)dxdyg(x,DDf(x,y)0,D1f(x,y)dxdyf(x,1}例1.Dxy|1}例1.Dxy|x2(x2y2)2(x2y2)cos(x2y2)2cos(x2y2)I3I2I1I1I3例例例4（03一Dx例4（03一Dxy|0x0y}證明Iesinesinydxdy2Desinydxdyesinesin2IesinDD4dxdy4D得題型六：二重積分證一元積分不bdf(x)g(題型六：二重積分證一元積分不bdf(x)g(acbdf(x)dxg(acbdf(x)dxg(x)dxacbb1.證明例f(x)dx]2(bf2(x)dx(a[aabbb解fbb1.證明例f(x)dx]2(bf2(x)dx(a[aabbb解f[f(x)dxfaaabbf(x)dxf(aaf(x)f(Dbbbf2(x)dx(b (x)dx2(x)dxdy2aaaDf2(D2f(x)f(2f(x)f(y)dxdyf2(x)f2(DDf(x)f(y)dxdy12(x)f2(2DD2(x)dxdyD得例(x)為[0,1]上的單減且恒大于0f112(x)dxf2(x)dx連續(xù)函數(shù)，證明例(x)為[0,1]上的單減且恒大于0f112(x)dxf2(x)dx連續(xù)函數(shù)，證明 11xf(x)dxf(x)dx00待證1111Ixf(x)dxf(x)dx22f(x)dx (x)dx0000If2(x)yf(y)dxdyxf2(x)f(DDf2(x)f(y)(yDf2(x)f(f2(x)f(y)(yD12 (x)f(y)(yx)dxdyf2(y)f(x)(xy)dxdy]2[DD1[f(x)f(y)(yx)[f(x)f(2D例(x)在[a,b]上連續(xù)，且恒大于f fbb證明dx(例(x)在[a,b]上連續(xù)，且恒大于f fbb證明dx(b2faa證 fff(bbdxfaaD1f(x)f(y)2f(fDdxdy(bD例f(x)在[0,1]上連續(xù)12111f(y)dy證明f(x)dx]2f[例f(x)在[0,1]上連續(xù)12111f(y)dy證明f(x)dx]2f[0x0證11f(y)dyf(x)f(f(x)dx0x1f(x)f(f(x)f(2DD21212111f(y)dyf(x)dx]2f(x)dx[000二重積分綜合問(wèn)例1.（11一f(1,y0,f(x,1)0,f(xy)dxdyDDxy|0二重積分綜合問(wèn)例1.（11一f(1,y0,f(x,1)0,f(xy)dxdyDDxy|0x1,0yIxyf//(x,D10101//(x,y)dy 1解Iydf/(x,x001x01yf/(x,1f/(x,y)dyx0x01111//f(x,f(x,xx00001I 11f/(x,y)dy 1/xf(x,xx00001111(x,y)dx1I 11f/(x,y)dy 1/xf(x,xx00001111(x,y)dx/xdf(x,0000f(x,111[xf(x,000f(