微积分教学资料——chapter15.ppt
《微积分教学资料——chapter15.ppt》由会员分享,可在线阅读,更多相关《微积分教学资料——chapter15.ppt(95页珍藏版)》请在三一文库上搜索。
1、Chapter 15 Multiple Integrals 15.1 Double Integrals over Rectangles 15.2 Iterated Integrals 15.3 Double Integrals over General Regions 15.4 Double Integrals in polar coordinates 15.5* Applications of Double Integrals 15.6* Surface Area,15.1 Double Integrals over Rectangles,Volumes and Double Integra
2、ls,A function f of two variables defined on a closed rectangle,and we suppose that,The graph of f is a surface with equation,Let S be the solid that lies above R and under the graph of f ,that is ,(See Figure 1)Find the volume of S,Figure 1,1) Partition:,The first step is to divide the rectangle R i
3、nto subrectangles.,Each with area,2) Approximation:,A thin rectangular box:,Base:,Height:,We can approximate by,3) Sum:,4) Limit:,A double Riemann sum,Definition The double integral of f over the rectangle R is,if this limit exists.,The sufficient condition of integrability:,is integral on R,Theorem
4、1.,Theorem2.,and f is discontinuous only on a finite number of smooth curves,is integral on,Note,If then the volume V of the solid that,lies above that the surface is,Example 1,If,evaluate the integral,Solution,15.2 Iterated Integrals,Partial integration with respect to y defines a function of x:,We
5、 integrate A with respect to x from x=a to x=b, we get,The integral on the right side is called an iterated integral and is denoted by,Thus,Similarly,Fubinis theorem If f is continuous on the rectangle,then,More generally, this is true that we assume that f is bounded on R , f is discontinuous only
6、on a finite number of smooth curves, and the iterated integrals exist.,The proof of Fubinis theorem is too difficult to include In our class.,If f (x,y) 0,then we can interpret the double integral,as the volume V of the solid S that lies above,R and under the surface z=f(x, y).,So,Or,Example,Solutio
7、n,Example,Solution,Example,Solution,Specially,If,Then,Some examples of type I,Some examples of type II,Example 4,Find the volume of the solid enclosed,by the paraboloid and the planes,Solution,and above region,The solid lies under the paraboloid,So the volume is,Suppose that D is a bounded region, t
8、he double integral of f over D is,15.3 Double Integrals over General Regions,Suppose that D is a bounded region which can be enclosed in a rectangular region R.,A new function F with domain R:,If the integral of F exists over R, then we define the double integral of f over D by,Some examples of type
9、 I,Evaluate where D is a region of type I,A new function F with domain R:,If f is continuous on type I region D such that,then,Some examples of type II,If f is continuous on type II region D such that,then,Example 1,Solution,Type I,Type II,Properties of Double Integral Suppose that functions f and g
10、 are continuous on a bounded closed region D. Property 1 The double integral of the sum (or difference) of two functions exists and is equal to the sum (or difference) of their double integrals, that is, Property 2 Property 3 where D is divided into two regions D1 and D2 and the area of D1 D2 is 0.,
11、Property 4 If f(x, y) 0 for every (x, y) D, then Property 5 If f(x, y)g(x, y) for every (x, y) D, then Moreover, since it follows from Property 5 that hence,where S is the area of D.,Property 6,Property 7 Suppose that M and m are respectively the maximum and minimum values of function f on D, then w
12、here S is the area of D. Property 8 (The Mean Value Theorem for Double Integral) If f(x, y) is continuous on D, then there exists at least a point (,) in D such that where S is the area of D. f (,) is called the average Value of f on D,Example 2,Solution,Type II,Type I,Example 3,Solution,Type I,Type
13、 II,Example change the order of integration,solution:,We have,An alternative description of D is,Example change the order of integration,solution:,Example Prove that,Solution,An alternative description of D is,where,We have,So,Chapter 10 Parametric Equations and Polar Coordinates 10.3 Polar coordina
14、tes,10.3 Polar coordinates,The point o is called the pole The point P is represented by the ordered pair (r, ) and r, are call- ed polar coordinates of P is positive if measure in the cou- nterclockwise direction from the po- lar axis and negative in the clockwi- se direction,The connection between
15、polar and Cartesian coordinates,Example Convert the point from polar to Cartesian coordinates,Solution,Example Represent the point with Cartesian coordinates,Solution,in terms of polar coordinates.,Example Identify the curve by finding a Cartesian equation for the curve,Solution,Example Find a polar
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- 微积分 教学 资料 chapter15
链接地址:https://www.31doc.com/p-3056871.html