重积分在某些极限问题和定积分问题中的应用

Application of Multiple Integral in Some Limit Problems and Definite Integral Problems

摘要: 重积分是数学分析的重要组成部分，它与定积分类似，也是某种特定形式的和的极限。它在几何、物理、工程学等多个学科都有广泛的应用，并起着至关重要的作用。对这部分内容，现有的教材侧重讲解如何将重积分化为累次积分来计算，较少涉及如何将两个定积分的乘积转化为重积分。本文着重介绍两方面内容：一是极限与重积分结合的计算问题；二是某些特殊结构的定积分向重积分的转化问题。尽管这些问题在平常的教学过程中不常见，但却备受竞赛和考研这两大群体的青睐，其重要性毋庸置疑。这两部分内容要求学生综合运用所学知识分析问题、解决问题，这有助于培养他们的能力素质和数学认知结构，对学生具有一定的挑战性。

Abstract: Multiple integral is an important part of mathematical analysis. Similar to definite integral, it is also the limit of sum with a specific form. It is widely used in geometry, physics, engineering and other disciplines, and it plays a vital role. For this part, the current materials focus on how to convert multiple integral into iterated integral, and less on how to convert the product of two definite integrals into double integral. There are two aspects in this paper: one is the calculation problem of the combination of limit and multiple integral; the second is the transformation from definite integral with some special structures to multiple integral. Although these problems are not com-mon in the ordinary teaching process, they are favored by the two groups of competition and postgraduate entrance examination, and there is no doubt about their importance. The two aspects require students to comprehensively use their knowledge, analyze problems and solve problems, which is helpful to cultivate their ability and mathematical cognitive structure. It is challenging for students.