多重积分的解法.doc

山西师范大学现代文理学院本科毕业论文 浅谈多重积分的解法 姓 名 郭芳 系 别 数计系 专 业 数学与应用数学 班 级 1001 学 号 1090110138 指导教师 赵连阔 答辩日期 成 绩 浅谈多重积分的解法 内容摘要 积分学是微积分学的一个分支，经历了一个漫长的时期，广泛地应用于数学与物理中。在数学中主要是对几何图形的面积、体积等进行深入的研究，具有极限的基本性质；在物理学与工程技术中，如求非均匀平面的质量、重心、转动惯量等等，都是所要本课题所要讨论的重积分的实际背景。在高等数学中占有重要的地位，它不仅是教学中的重点，更是难点。 本课题的着重点是关于多重积分转化为累次积分时涉及到的积分区域确立的问题，即定限问题，以及积分次序的变化。在解决具体问题时，关键在于积分区域的确立。确立积分区域后就可以将多重积分转化为累次积分，进一步化解为定积分，最终得出结果。在转化的过程中，通常采用数形结合的思想，对其求解。 通过对国内外有关的学术刊物和相关著作等文献进行分析，并对其进行整理、分析、提炼，从二重积分和三重积分两个方面进行论述被积函数与被积区间之间的关系，并通过一些例题来进行具体说明。 关键词 多重积分 积分区域 积分次序 数形结合 Introduction to multiple integral method Abstract Integral calculus is a branch of calculus, has experienced a long period, widely used in mathematics and physics. In mathematics is mainly on the geometry of area, volume and so on in-depth research, with basic properties of the limit; In physics and engineering technology, such as the quality of the inhomogeneous plane, center of gravity, inertia, etc., are to this topic to be discussed by the repeated integral actual background. Occupies an important position in the higher mathematics, it is the key to teaching, not only is more difficult. About this topic with the focus on the multiple integral is transformed into the iterated integral involve the issue of integral area established, which set the limit problem, and the change of the integral order. When solving specific problems, the key lies in the establishment of the region. After establishing region can convert multiple integral to the iterated integral, further dissolve into definite integral, the final results. In the process of transformation, usually adopts the number form combining ideas, to solve. Based on the relevant academic journals and related works of literature at home and abroad is analyzed, and carries on the sorting, analysis, extraction, from two aspects of double integral and triple integrals of integrand and be discussed the relationship between t