[理学]线性代数III-chapter32.ppt

Example 9 Solution Because both the integrand and the domain of integration are more conveniently expressed in polar coordinates, we compute the integral by means of formula above. Example 10 Solution Example 11 Solution 4、 Computing double integrals using the symmetry Solution Example 12 Example 13 Solution Example 14 Solution * Prof Liubiyu Contents §3.2 Computation of double integrals §3.3 Application of double integrals §3.1 Concepts and properties of double integrals §3.4 Concept of triple integral and the computation in rectangular coordinates §3.6 Integration by substitution for multiple integrals §3.7 Application of triple integrals §3.5 Computation of triple integrals in cylindrical and spherical coordinates §3.2 Computation of double integrals In this section, we will first show how to use the rectangular coordinates to evaluate the double integrals. This method requires that both the integral domain and integrand be described in rectangular coordinates. Then we will show how to evaluate the double integral using polar coordinates. This method is especially appropriate when the integral domain has a simple description in polar coordinates, for instance, if it is a disk or cardioid. Now we discuss the computation of double integrals in term of the geometric meaning of the double integral. 1、 Computation methods for double integrals in rectangular coordinates Notation 1: The computation of the double integral is reduced to the computation of two definite integrals of the function of one variable one after the other. Notations Example 1 Solution Example 2 Solution Example 3 Solution Notation 5: It can be seen from the above examples that when we reduce a double integral to an iterated integral, proper choice of the order of the integration is important Example 4 Solution 2、 Interchanging the order of integration It is often useful to make such a change when evaluating iterated integrals, sinc