摘 要
在中学我们就初步学习了关于对称性的知识，然后是逐步深入学习，到大学的学习中，我们接触到了微积分这个概念，我们发现，积分知识在微积分的学习中即是重点也是难点，尤其是在解决积分计算的问题上。本文介绍了几种常见的对称性在积分计算过程中的几个定理和证明，经过多方考察和查阅资料，运用分类对比和归纳总结的数学思想方法，研究对称性在定积分、重积分、曲线积分和曲面积分的应用。首先是通过了解对称性的研究背景和研究意义，引出对称性定理；在研究对称性应用时，需要理解这些积分的一般定义，本人在查找资料和学习理解之后，合理引入对称性定理，并对同类定理进行了证明，归纳总结它们的对称性特点，分类讨论，然后根据定理，举例子说明或解决一些积分计算问题。因此，在遇到一些具有对称性的函数或积分时，可以利用对称性定理来减少数值的计算量，使计算过程得以简化。
关键词：对称性；奇偶性；定积分；重积分；曲线积分；曲面积分
The application of symmetry in integral
Abstract
In the middle school we have initially learned about the symmetry of knowledge, and then gradually in-depth study, to the University of learning, we have access to the concept of calculus, we found that integral calculus in the study is the focus is also difficult, especially in solving the problem of integral computing. This paper studies the application of symmetry in definite integral, heavy integral, curvilinear integral and curved area division by means of various investigation and reference data, using the mathematical method of classification comparison and summarization. The first is to understand the research background and significance of symmetry, and then draw the symmetry theorem. In the study of symmetric applications, first of all, we must understand the general definition of these integrals, after looking at the data and learning to understand, the rational introduction of the symmetry theorem, and the proof of the same kind of theorem, summed up their symmetry characteristics, classified discussion, and then according to the theorem, examples to solve the problem Therefore, in the face of some symmetric functions or integrals, the symmetry theorem can be used to reduce the numerical calculation and optimize the integral calculation.
Key words: Symmetry; parity; definite integral; weight integral; curvilinear integral; curved area
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