Double integrals and their applications

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 $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}=\frac{{\pi }^2}{6}$.