TY  - JOUR
AU  - ZHAO, Zhengang
PY  - 2024
TI  - Double integrals and their applications
JO  - Journal of Capital Normal University (Natural Science Edition)
SN  - 1004-9398
SP  - 68
EP  - 76
VL  - 45
IS  - 1
AB  - 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                      ∞                  1                  n                    2              =                              π                            2              6  .
UR  - https://doi.org/10.19789/j.1004-9398.2024.01.009
DO  - 10.19789/j.1004-9398.2024.01.009