第九章重积分.doc

第九章 重积分Chapter 9 Multiple Integrals9.1 二重积分的概念与性质 (The Concept of Double Integrals and Its Properties)一、二重积分的概念 (Double Integrals)定义 ( 二重积分的定义 ) 设 D 是平面的有界闭区域 ,是定义在 D 上的函数。将 D 任意分成 n 个小区域,它们的面 积用 表示。在每个上任取一点，并作和。假设存在一个确定的数满足：任给，存在，使得当各小区域的直径中的最大值小于时，就有不管区域D的分法如何，的取法如何。这样就称f在D上可积，I称为f在D上的二重积分，记作或Definition (The Double Integral) Let D be a bounded closed region in the 巧 1 plane and f a function defined on D. Partition D arbitrarily into subregions ，whose area is denoted by Choose arbitrarily a point in and then form the sum 。Supposethat there exists a fixed number I such that for any , there exists a such that if the length of the longest diameter of those subregions in a partition of D is less than , then，no matter how the partition is and how those points are chosen from Then is said to be integrable over Dand I is the double integral of over D ,written ,or 二、二重积分的性质 (Properties of Double Integrals)性质 1 两个函数和 ( 或差 ) 的二重积分等于它们二重积分的和 ( 或差 ), 即.Property 1 The double integral of the sum(or difference) of two functions is equal to the sum( or difference) of their double integrals, that is性质 2 被积函数前面的常数因子可以提到积分号前面 , 即,若k为常数。Property 2 The constant factor in the integrand function can be taken out of the double integral,that is ,if k is a constant.性质 3 二重积分关于积分区域具有可加性 , 即如果被 分成两个区域 和,的面积为 0, 则有.Property 3 The double integral is additive with respect to the integration region, that is, if is divided into two regions and and the area of is 0, then性质 4 若对任意，有，则有。Property 4 If for every ,then .性质5 若对任意, 有，则。Property 5 If for every ,then .性质 6 假设 M 和 m 分别是函数 f 在上的最大值和最小 值 , 则，其中是区域的面积。Property 6 Suppose that M and m are respectively the maximum and minimum values of function f on , then,where is the area of .性质 7( 二重积分的中值定理 ) 若在闭区域上连续 , 则在上至少存在一点使得，其中是区域的面积Property 7 (The Mean Value Theorem for Double Integral)If is continuous on the closed region , then there exists at least a point in such that , where is the area of .9.2 二重积分的计算法 (Evaluation of Double Integrals)一、直角坐标的二重积分 (Double Integrals in Rectangular Coordinates)定理 设 在xy平面上的有界闭区域 D 上连续 ,(1)若区域由和所给出，其中是上的连续函数，则。(2) 若区域由和所给出，其中是上的连续函数，则。Theorem Let be continuous on a bounded closed region in the -plane.(1) If is given by and , where are continuous functions of on, then(2) If is given by and , where are continuous functions of on , then二 极坐标下的二重积分 (Double Integrals in Polar Coordinates)如果积分区域由极坐标形式给出，则。If the region is given by in polar coordinates, then。9.3 三重积分 (Triple Integrals)一、三重积分的概念 (Triple Integrals)定义 设是定义在空间有界闭区域上的三元函数。如果存在一个确定的数, 满足 :任给, 存在, 对 的任 意一个分法 , 设小区域为，它们的体积用表示 , 在每个中任取一点，当小区域的最大直径小于时，不等式成立，则成为在D上的三重积分，记作,或.Definition Suppose that is a continuous function of three variables defined on a bounded closed region in space. If there is a number such that for any , there exists a such that, for any partition to with subregions , whose volume are denoted by and any pointschosen arbitrarily from respectively, the inequality holds true whenever the diameter of the largest subregion is less than, then A is said to be the triple integral of function over the region ，written，or 二、三重积分的计算 (Evaluation of Triple Integrals)1. 利用直角坐标 (Rectangular Coordinates) 计算三重积分 设是定义在空间有界闭区域上的连续的二元函数,由给出，则Suppose that is a continuous function of three variables defined on a bounded closed region in space given by.Then2. 利用柱面坐标 (Cylindrical Coordinates) 计算三重积分 当空间立体区域有一个对称轴时 , 计算上的三重积分通常使用柱坐标比较容易.When a solid region in three-space has an axis of symmetry, the evaluation of triple integrals overis often facilitated by using cylindrical coordinates.柱坐标和笛卡尔 ( 直角 ) 坐标之间的关系为.Cylindrical and Cartesian (rectangular) coordinates are related by the equations .设是一个型立体区域 , 它在平面的投影是型的。 如果在上连续 , 则.Let be a z-sample solid and suppose that its projection in the -plane is r-sample. If is continuous on , then.3. 利用球面坐标 (Spherical Coordinates) 计算三重积分 当空间立体区域关于某个点对称时,计算上的三重积分通常使用球坐标比较容易。When a solid regionin three-space is symmetric with respect to a point, the evaluation of triple integrals over is often facilitated by using spherical coordinates.球坐标和笛卡尔 ( 直角 ) 坐标之间的关系为:The equations relate sphericalcoordinates and Cartesian (rectangular) coordinates.球坐标中的体积元素由给出，因此有 The volume element in spherical coordinates is given by hence we have4. 重积分中的变量代换 (Change of Variables in Multiple Integrals)假设建立了旧变量与新变量的关系。定义一个函数，叫做雅可比行列式。这样 , 在适当的限制条件下 , 就有。Suppose that relate the old variablesandto new variables u and v. Define a function ，called the Jacobian，by 。Then，under suitable conditions on the functions and and with appropriate limits on the integral signs，对于三元情形 , 其中，则雅克比行列式由式子给出，同样在变换后的积分会出现因子。In the three-variable case, where ，the Jacobian is defined by Again， is the extra factor that appears in the transformed integral。9.4 重积分的应用 (Applications of Multiple Integrals)设曲面 S 由方程给出 ,D 为曲面 S 在平面上的投影区域 , 函数在 D 上具有一阶连续偏导数和。则 S 的表 面积为。Suppose the surface S is defined by the equation 。 D is the projection region of S in the-plane. Assume that has continuous first partial derivatives and on D. Then the furface area of S is g