This paper begins by defining the double integral over a rectangular region,which is divided into subsections by using a grid of lines parallel to the x-axis and y-axis.We provide detailed and complete proofs for the related properties of double integrals.Furthermore,we demonstrate that a necessary and sufficient condition for a bounded function to be integrable over a rectangular region is that the set of its points of discontinuity within the region forms a set of measure zero.Assuming that a function is Riemann integrable over a rectangular region,we successfully overcome the problem where the Riemann integral may not exist when fixing variable x and considering the function as that of y over the corresponding closed interval.Subsequently,we establish the integration of a bounded function over a bounded set and convert the integration over a bounded set into an integration over a rectangular area.Then,we discuss the applications of the double integral,prove that if two mixed partial derivatives of a binary function defined on an open set are continuous,they must be equal.Lastly,the double integral is employed to validate the summation formula for the series ∞∑n=1/n2=π2/6.
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