《9.7重积分习题课》由会员分享，可在线阅读，更多相关《9.7重积分习题课（48页珍藏版）》请在金锄头文库上搜索。
1、重积分复习 一、知识点回顾； 二、典型例题分析； 三、练习Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.积分区域为：二重积分 （几何意义：曲顶柱体体积）物理意义：平面的质量X型 Eval
2、uation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.积分区域为：Y型 Evaluation only.Evaluation only. Created with Aspose.Slides for .
3、NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.二重积分在极坐标下的计算公式1、被积分函数或者积分区域含有圆，常用极坐标；2、在积分中注意使用对称性与奇偶性;3、一般变量代换中注意雅可比行列式要加绝对值1 1、极点不在区域 D 的内部2 2、极点位于区域 D 的内部3 3、极点在区域 D 的边界上Evaluation
4、 only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.二重积分的计算方法是累次积分法，化二重 积分为累次积分的步骤是：作出积分区域的草图选择适当的坐标系选定积分次序，定出积分限1、关于坐标系的选择：这要从积分区域的
5、形状和被积函数的特点 两个方面来考虑Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.被积函数呈 常用极坐标其它以直角坐标为宜2、关于积分次序的选择选序原则能积分，少分片，计算简单3、关于积
6、分限的确定二重积分的面积元 为正确定积分限时一定要保证下限小于上限积分区域为圆形、扇形、圆环形极坐标系下勿忘 rEvaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.4、关于对称性利用对称性来简
7、化重积分的计算是十分有效的， 它与利用奇偶性来简化定积分的计算是一样的，不 过重积分的情况比较复杂，在运用对称性时要兼顾 被积分函数和积分区域两个方面，不可误用对若D关于 x 轴对称Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004
8、-2011 Aspose Pty Ltd.若D关于 y 轴对称若D关于原点对称Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.称为关于积分变量的轮换对称性 是多元积分所独有的性质奇函数关于
9、对称域的积分等于0，偶函数关于 对称域的积分等于对称的部分区域上积分的两倍 ，完全类似于 对称区间上奇偶函数的定积分的性 质、简单地说就是:若 D 关于直线 y = x 对称Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-201
10、1 Aspose Pty Ltd.例1解1解2注意两种积分次序的计算效果！Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.解D关于 x , y 轴及原点及 y = x 对称故故例2 计算E
11、valuation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.解例3 计算D1D2Evaluation only.Evaluation only. Created with Aspose.Slides f
12、or .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.解D的边界 极点在D的边界上 圆周在(0, 0)的切线斜率为故例4 计算Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0
13、.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.例5 计算D2D1解Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.
14、0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.例6设 f (x) 在 0,1 上连续求解DEvaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyrigh
15、t 2004-2011 Aspose Pty Ltd.试将二重积分 化成定积分解由积分域和被积函数的对称性有用极坐标例7Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.为将二次积分化为所需要的定积分 ，须变换积分次序 DD1Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.即：化为一个定积分和一个二重积分的运算即：化为一个定积分和一个二重积分的运算三重积
