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1、Chapter 15 Multiple Integrals15.1 Double Integrals over Rectangles15.2 Iterated Integrals15.3 Double Integrals over General Regions15.4 Double Integrals in polar coordinates15.5* Applications of Double Integrals15.6* Surface Area15.7 Triple Integrals15.1 Double Integrals over RectanglesVolumes and D

2、ouble IntegralsA function f of two variables defind on a closed rectangleand we suppose that The graph of f is a surface with equation Let S be the solid that lies above R and under the graphof f ,that is ,(See Figure 1)Find the volume of SFigure 11) Partition:The first step is to divide the rectang

3、le R into subrectangles.Each with area2) Approximation:A thin rectangular box:Base:Hight:We can approximate by3) Sum:4) Limit:A double Riemann sumDefinition The double integral of f over the rectangleR isif this limit exists.The sufficient condition of integrability:is integral on RTheorem1.Theorem2

4、.and f is discontinuous only on a finite number ofsmooth curvesis integral on Example 1If evaluate the integralSolution15.2 Iterated IntegralsPartial integration with respect to y defines a functionof x:We integrate A with respect to x from x=a to x=b,we getThe integral on the right side is called a

5、n iterated integraland is denoted by Thus Similarly Fubinis theorem If f is continuous on the rectanglethen More generally, this is true that we assume that f is boundedon R , f is discontinuous only on a finite number of smooth curves, and the iterated integrals exist.The proof of Fubinis theorem i

6、s too difficult to includeIn our class.If f (x,y) 0,then we can interpret the double integralas the volume V of the solid S that lies above R and under the surface z=f(x, y).SoOr Example 1Solution 1Solution 2Example 2SolutionExample 3SolutionSpecially If Then 15.3 Double Integrals over General Regio

7、nsSuppose that D is a bounded region which can be enclosed in a rectangular region R.A new function F with domain R:0D0DIf the integral of F exists over R, then we define the double integral of f over D byIf the integral of F exists over R, then we define the double integral of f over D bySome examp

8、les of type IA plane region D is said to be of type I if Where and are continuous on a,bEvaluate where D is a region of type IA new function F with domain R:If f is continuous on type I region D such that then Some examples of type IIA plane region D is said to be of type II if Where and are continu

9、ous on a,bIf f is continuous on type II region D such that then Example 1 Evaluate ,where D is the regionbounded by the parabolas SolutionType I Type IIProperties of Double IntegralSuppose that functions f and g are continuous on a bounded closed region D.Property 1 The double integral of the sum (o

10、r 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.Property 4 If f(x, y) 0 for every (x, y) D, thenProperty 5 If f(x, y)g(x, y) for every (x, y

11、) D, thenMoreover, since it follows from Property 5 that hence Property 6 Suppose that M and m are respectively the maximum and minimum values of function f on D, then where S is the area of D.Property 7 (The Mean Value Theorem for Double Integral)If f(x, y) is continuous on D, then there exists at

12、least a point (,) in D such that where S is the area of D. f (,) is called the average Value of f on DExample 2 Evaluate ,where D is the regionbounded by the parabolas SolutionType II Type IExample 3 Evaluate ,where D is the regionbounded by the parabolas SolutionType IExample 4 change the order of

13、integration solution：xyo231xyoxyoExample 5 change the order of integration solution：15.4 Double Integrals in polar coordinatesA polar rectangle whereThe “center” of the polar subrectangle has polar coordinates The area of is Change of polar coordinate in a double integralIf f is continuous on a pola

14、r rectangle R given by , ,where , then 1.If f is continuous on a polar region of the form then 2.If f is continuous on a polar region of the form then 3.If f is continuous on a polar region of the form then Example 1 SolutionEvaluate ,where D is the regionbounded by the circles 1212oExample 2 Evalua

15、te where D = Solution In polar coordinates, the equationx2 + y2 2x = 0 es The disc D is given by D:Therefore 15.7 Triple IntegralsLets first deal with the simplest case where f is defined on a rectangular box:The first step is to divide B into lmn sub-boxesEach sub-box has volume Then we form the tr

16、iple Riemann sumDefinition The triple integral of f over the box B isif this limit exists.Fubinis Theorem for Triple Integrals If f is continuous on the rectangular box then Example 1 Evaluate the triple integral where B is given by Solution Type 1 In particular, if the projection D of E onto the xy

17、-plane is a type I plane region, then If D is a type II plane region, thenExample 2 Evaluate where E is the solid tetrahedron bounded by the four planesand Solution The projection of E 0n the xy-plane is the triangular region, and we haveA solid region E is of type 2 if it is of the formA solid regi

18、on E is of type 3 if it is of the formExample 3 Evaluate where E is the region bounded by the surfaces and the plane Solution The projection of E onto the xz-plane is the circle , and we have15.9 Change of Variables in Multiple IntegralsDefinition The Jacobian of the transformation T given by x=g(u,v) and y=h(u,v) is Change of variables in a Double Integral The transformation T is given by x=g(u,v) and y=h(u,v). Then For instance,Thus,Example 1 Evaluate the integral where D is the region