二重积分中值定理中间点的进一步讨论

众所周知,二重积分中值定理一般表述为：“设Ｄ为平面上有界闭区域,ｆ（ｘ,ｙ）在Ｄ上连续,ｇ（ｘ,ｙ）在Ｄ上可积且不变号,则至少存在一点（ζη）∈Ｄ,使得∫∫Ｄｆ（ｘ,ｙ）ｇ（ｘ,ｙ）ｄｘｄｙ＝ｆ（ζη）∫∫Ｄｇ（ｘ,ｙ）ｄｘｄｙ。”进一步证明了中间点（ζη）可选为Ｄ的内点,其结论与方法完全可推广到一般多重积分中值定理。

A Further Study:the Mean Value Theorem in Twofold Integrals

it is Well-known that the mean value theorem in twofold integrals is stated as:"Let D be a bounded closed region of R2, f(x,y) a continuous function and g(x,y) an integrable function with invariant sign on D, then exits at least a point such that 'This paper goes further to prove that the point can be an interior point within D. And such a conclusion and the method employed to prove it are completely applicable to mean value theorem in general multiple integrals.