微積分英文版教學課件：Chapter 15 Multiple Integrals

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 Area15.1 Double Integrals over RectanglesVolumes and Double IntegralsA func

2、tion f of two variables defined 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 rectangle R into subrectang

3、les.Each with area2) Approximation:A thin rectangular box:Base:Height: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.and f is discontin

4、uous only on a finite number ofsmooth curvesis integral on NoteIf then the volume V of the solid thatlies above that the surface is 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 t

5、o x=b,we getThe integral on the right side is called an 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 i

6、terated integrals exist.The proof of Fubinis theorem is 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 ExampleSolutionExample SolutionExample SolutionSpecially If Then So

7、me examples of type IA plane region D is said to be of type I if Where and are continuous on a,bSome examples of type IIA plane region D is said to be of type II if Where and are continuous on c,dExample 4 Find the volume of the solid enclosedby the paraboloid and the planesSolutionand above regionT

8、he solid lies under the paraboloidSo the volume isSuppose that D is a bounded region, the double integral of f over D is15.3 Double Integrals over General RegionsSuppose 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

9、 exists over R, then we define the double integral of f over D bySome examples 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 o

10、f type IIA plane region D is said to be of type II if Where and are continuous on c,dIf 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 continuou

11、s 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.Property 4 If f(x

12、, y) 0 for every (x, y) D, thenProperty 5 If f(x, y)g(x, y) for every (x, y) D, thenMoreover, since it follows from Property 5 that hence where S is the area of D.Property 6Property 7 Suppose that M and m are respectively the maximum and minimum values of function f on D, then where S is the area of

13、 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 DExample 2 Evaluate ,where D is the regionbounded by the parabolas SolutionType II Type I

14、Example 3 Evaluate ,where D is the regionbounded by the parabolas SolutionType IType IIExample change the order of integration solution：We haveAn alternative description of D isxyo231xyoxyoExample change the order of integration solution：Example Prove thatSolutionAn alternative description of D iswh

15、ere We haveSoChapter 10 Parametric Equations andPolar Coordinates 10.3 Polar coordinates10.3 Polar coordinates0Polar axisP(r, )rThe point o is called the poleThe point P is represented by theordered pair (r, ) and r, are call-ed polar coordinates of Pis positive if measure in the cou-nterclockwise d

16、irection from the po-lar axis and negative in the clockwi-se directionThe connection between polar and Cartesian coordinatesyrP(x,y)=p(r,)Example Convert the point from polar to CartesiancoordinatesSolutionExample Represent the point with Cartesian coordinatesSolutionin terms of polar coordinates.Ex

17、ample Identify the curve by finding a Cartesian equationfor the curve SolutionExample Find a polar equation for the curve representedby the given Cartesian equation Solution1212o2oChapter 15 Multiple Integrals15.1 Double Integrals over Rectangles15.2 Iterated Integrals15.3 Double Integrals over Gene

18、ral Regions15.4 Double Integrals in polar coordinates15.5* Applications of Double Integrals15.6* Surface Area15.4 Double Integrals in polar coordinatesA polar rectangle whereThe “center” of the polar subrectangle has polar coordinates The area of is Change to polar coordinate in a double integralIf

19、f is continuous on a polar rectangle R given by , ,where , then SolutionExample 1 Evaluate ,where D is the regionin the upper half plane bounded by the circles 1212o1.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

20、 a polar region of the form then Solution：Example Find ，。D is given bySoExample Evaluate，Solution2oWe haveImproper Integral (over the entire plane)where is the disk with radius and center the origin.where is the square with vertics .DExample15.7 Triple Integrals f is defind on the rectanglar box BDe

21、finition The triple integral of f over the boxB isif this limit exists.is integral on BTheorem1.Fubinis theorem If f is continuous on the rectanglar then box BExampleEvaluate the triple integralWhere B is the rectangular box given bySolutionThe triple integral over a general bounded region Ein three

22、 -dimensional space A solid region E is said to be of type I ifthenIf the projection D of E onto the xy-plane is a type I plane regionthenthenIf the projection D of E onto the xy-plane is a type II plane regionExampleEvaluate ,where E is the solid tetrahedron bounded by the four planes andtetrhi:drn

23、 Solution We have it is a type I region. 15.8 Triple Integrals in Cylindrical and Spherical Coodinates1.Cylindrical CoodinatesIfwhere D is given in polar coordinates by thenExampleA solid E lies within the cylinderbelow the planeSolutionabove the paraboloidThe density at any point is proportionalto

24、its distance from the axis of the cylinder.Find the mass of E.We haveSince the density at any point is proportionalto its distance from the z-axis, the density function iswhere k is the proportionality constant. Therefore,the mass isExampleEvaluate the integral by changing to cylindrical coordinates

25、.SolutionThe solid region has a much simpler description in cylindrical coordinates:The solid region is2.Spherical Coodinateswhere E is a spherical wedge given byIf E is a general spherical region such asthenExampleEvaluate the integral by changing to spherical coordinates.SolutionThe solid region h

26、as a much simpler description in spherical coordinates:The solid region isExampleUse spherical coordinates to find the volumeSolutionThe volume of E isThe solid isof the solid that lies above the cone andbelow the sphere15.9 Change of Variables in Mutiple IntegralsA transformation T from the uv-plan

27、e to the xy-planewhereorT is a transformation, which means that and have continuous first-order partial derivatives.A transformation T is a function whose doman and range are both subsets of If then the point is called the image of the point If no two points have the same image,T is called one-to-on

28、e. A transformation T on a region S in the uv-plane.T transforms S into a region R in the xy-plane called the image of S,consisting of the images of all points in S.If T is one-to-one transformation,then it has an inverse transformation from the xy-plane to the uv-plane.ExampleA transformation is defined byS is the triangular region with verticesSolutionTh