解答题
2.已知函数f(x，y)具有二阶连续偏导数，且f(1，y)=0，f(x，1)=0，
f(x，y)dxdy=a，其中D={(x，y)|0≤x≤1，0≤y≤1}，计算二重积分I=
f(x，y)dxdy=a，其中D={(x，y)|0≤x≤1，0≤y≤1}，计算二重积分I=
【正确答案】因为f(1，y)=0，f(x，1)=0，所以fy'(1，y)=0，fx'(x，1)=0．
从而
I=∫01xdx∫01yf(x，y)dy=∫01x[yfx'(x，y)∫01一fx'(x，y)dy]dx
=一∫01dy∫01xfx'(x，y)dx=一∫01[xf(x，y)|x=0x=1一∫01f(x，y)dx]dy
=∫01dy∫01f(x，y)dx=a．
a=∫01dy∫01f(x，y)dx
=∫01[xf(x，y)|x=0x=1一∫01xfx'(x，y)dx]dy
=一∫01dx∫01xfx'(x，y)dy
=一∫01[xfx'(x，y)=|y=0y=1一∫01xyfxy"(x，y)dy]dx
=
从而
I=∫01xdx∫01yf(x，y)dy=∫01x[yfx'(x，y)∫01一fx'(x，y)dy]dx
=一∫01dy∫01xfx'(x，y)dx=一∫01[xf(x，y)|x=0x=1一∫01f(x，y)dx]dy
=∫01dy∫01f(x，y)dx=a．
a=∫01dy∫01f(x，y)dx
=∫01[xf(x，y)|x=0x=1一∫01xfx'(x，y)dx]dy
=一∫01dx∫01xfx'(x，y)dy
=一∫01[xfx'(x，y)=|y=0y=1一∫01xyfxy"(x，y)dy]dx
=
【答案解析】