重积分定限问题的探讨

论文编号:XXLW006论文字数:7704,页数:25

摘 要

重积分是多元函数积分学的一块重要内容,重积分的计算其思想是化重积分为累次积分,确定每一累次积分的上下限历来是教与学的一个重点与难点,本文较为详细地讨论了空间曲线,立体在坐标面上的投影,根据积分区域的不同情形,介绍了二重积分的具体定限方法;三重积分的定限法:图形定限法,平面图形与代数分析结合定限法和代数分析定限法。

关键词: 投影; 二重积分; 三重积分;累次积分; 积分限

Study on the Problem of Limit defination

Of multivariable Integral

Abstract

Multivariable Integrals is an important content of multivariable function integration.Its calculation thoughts is tun the multivariable integral into repeated integrals, to be definite the lpper limits and inferior limit are always both important and difficult during learning and teaching of multivariable Integrals.In this paper,we discuss the Projection at axise of space curves and cuble and based on different case of integral field.we introduce special methods to determine the limit of Double integral, the determined of Triple integrals ,graph determined mehods ,combine algebric analysis and plane gragh mehods.

Key words：:Projection;Double Integral;Triple Integral;Repeated Integral;Bounds of Integrals

目 录

中文摘要..................................................................... i

英文摘要.................................................................... ii

目录........................................................................iii

引 言........................................................................1

1. 空间曲线、立体在坐标面上的投影.............................................2

1.1.空间曲线在坐标面上的投影................................................2

1.1.1 空间曲线在面上的投影.........................2

1.1.2 空间曲线在面上的投影.......................2

1.2.体在坐标面上的投影......................................................3

1.2.1 单个曲面围成的体的投影............................................4

1.2.2 方程都含z坐标的两个曲面围成的体的投影............................4

1.2.3不含z坐标（柱面）和含z坐标的曲面（至多两个）围成的体的投影...........5

1.2.4 由两个同型曲面和另一个曲面所围的体的投影..........................6

2. 二重积分积分限的确定.......................................................7

2.1 在直角坐标系下的二重积分积分限的确定....................................7

2.1.1 为X—型区域....................................................7

2.1.2 为Y—型区域....................................................8

2.1.3 既非X—型，又非Y—型...........................................9

2.2 极坐标系下的二重积分积分限的确定.......................................11

3. 三重积分的积分限的确定....................................................13

3.1 利用立体图形确定积分限.................................................13

3.2平面图形与代数分析相结合确定积分限.....................................13

3.3 代数分析法确定三重积分的积分限......... ...............................19

4. 结束语....................................................................23

致谢.........................................................................24

参考文献.....................................................................25