高数 07.08 二重积分的计算

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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

,f(x,y)0D,DX
D:{φ1(x)yφ2(x)axb
Df(x,y)dxdy=ba[φ2(x)φ1(x)f(x,y)dy]dx
DY
D:{ψ1(y)xψ2(x)cyd
Df(x,y)dxdy=dc[ψ2(y)ψ1(y)f(x,y)dx]dy

:(1)XY,Df(x,y)dxdy=ba[φ2(x)φ1(x)f(x,y)dy]dx=dc[ψ2(y)ψ1(y)f(x,y)dx]dy便,.(2),XY.

1.I=Dxydσ,D线y=1,x=2y=x.
:1.DX,D:{1yx1x2
I=21dxx1xydy=21[12xy2]x1=21[12x312x]dx=[18x414x2]21=98
1.DY,D:{yx21y2
I=21dy2yxydx=21[12x2y|2y]dy=21[2y12y3]dy=[y218y4]21=98

2.Dxydσ,D线y2=x线y=x2.
:便,xy,
D:{y2xy+21y2
Dxydσ=21dyy2y+2xydx=21[12x2y]y+2y2dy=21[12(y+2)2y12y5]dy=21[12(y5+y3+4y2+4y)]dy=12[16y6+14y4+43y3+2y2]21=458

3.Dsinxxdxdy,D线y=x,y=0,x=π.
:
D:{0xπ0yx
Dsinxxdxdy=π0dxx0sinxxdy=π0sinxx[y]x0dx=π0sinxxxdx=π0sinxdx=[cosx]π0=2

4.I=20dxx220f(x,y)dy+222dx8x20f(x,y)dy
:y=1x2x2+y2=8(2,2)
D1:0x20y12x2
D2:{2x220y8x2
xy
D:{0y22yx8y2
I=20dy8y22yf(x,y)dx

,r=线θ=,DΔσk(k=1,2,,n),Δσk=12(rk+Δrk)2Δθk12r2kΔθk=12[rk+(rk+Δrk)]ΔrkΔθk=rk¯ΔrkΔθkΔσk(rk¯,θk¯),ξk=rk¯cosθk¯+rk¯sinθk¯limλ0nk=1f(ξk,ηk)Δσk=limλ0nk=1f(rk¯cosθk¯,rk¯sinθk¯)rk¯ΔrkΔθDf(x,y)dσ=Df(rcosθ,rsinθ)rdrdθ
D:{φ1(θ)rφ2(θ)αθβ
Df(rcosθ,rsinθ)rdrdθ=βαdθφ2(θ)φ1(θ)f(rcosθ,rsinθ)rdr

,D:{φ1(θ)rφ2(θ)0θ2π
Df(rcosθ,rsinθ)rdrdθ=2π0dθφ(θ)0f(rcosθ,rsinθ)rdr

f1,D
σ=Ddσ=122π0φ2(θ)dθ

:Dx,y,θ?
(1)r=φ(θ)x,θ[0,π]
(2)r=φ(θ)y,θ[π2,π2]

5.Dex2y2dxdy,D:x2+y2a2.
ex2y2,
:D:{0ra0θ2π
Dex2y2dxdy=Der2rdrdθ=2π0dθa0er2rdr=2π0[12er2]a0dθ=2π01ea22dθ=π(1ea2)

6.x2+y2+z24a2x2+y2=2ax(a>0)().
:D:0r2acosθ,0θπ2

V=4D4a2r2rdrdθ=4π20dθ2acosθ04a2r2rdr=323a3π20(1sin3θ)dθ=323a3(π223)

内容小结
(1)二重积分化为累次积分的方法
直角坐标系情形:
若积分区域为:
D={(x,y)|axb,y1(x)yy2(x)}
Df(x,y)dσ=badxy2(x)y1(x)f(x,y)dy

若积分区域为:
D={(x,y)|cyd,x1(y)xx2(y)}
Df(x,y)dσ=dcdyx2(y)x1(y)f(x,y)dx

极坐标系情形:
若积分区域为
D={(r,θ)|αθβ,φ1(θ)rφ2(θ)}
Df(x,y)dσ=Df(rcosθ,rsinθ)rdrdθ=βαdθφ2(θ)φ1(θ)f(rcosθ,rsinθ)rdr

(2)计算步骤及注意事项
先画出积分域
选择坐标系
确定积分序
写出积分限
计算要简便,充分利用对称性

练习
1.f(x)C[0,1],10f(x)dx=A,I=10dx1xf(x)f(y)dy
:,x,yI=10dx1xf(x)f(y)dy=10dxx0f(x)f(y)dy2I=10dx1xf(x)f(y)dy+10dyx0f(x)f(y)dy=10dx10f(x)f(y)dy=10f(x)dx10f(y)dy=A2I=A22

2.I=2a0dx2ax2axx2f(x,y)dy(a>0,
:y=2axx=y22ay=2axx2x=a±a2y2I=2a0dx2ax2axx2f(x,y)dy=10dyaa2y2y22f(x,y)dx+a0dy2aa+a2y2f(x,y)dx+2aa2ay22af(x,y)dx

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