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1、13.1 Double Integrals over RectanglesProblem IntroductionVolume=Bottom Area heightVolume=？),(yxfz=DThe volume of the curved top cylinderProblem IntroductionThe method of dividing,approximating,summing,and limitis used to obtain the volume of the curved top cylinder.Problem Introduction(2)The volume
2、of each smallcurvedtopcylinderisapproximately represented by aflat-top cylinder.xzyoD),(yxfz=i),(ii(1)Divide the bottom of the curved top cylinder and take a typical small area.),(lim10iiniifV =(4)Limit:the volumeFour Steps:(Geometrical meaning)(3)Sum the volume of all the small flat-top columns.Pro
3、blem IntroductionFind the Mass(physical meaning)i),(iiThere is a flat sheet that occupies the closed area D on the xOy plane.The surface density at the point(x,y)is ,assuming that is continuous on D.What is the mass of the flat sheet?(1)The flat sheet is divided into smallpieces.Take a typical small
4、 piece,(2)Approximate it as a uniform thin sheet.The mass of the flat sheet D isapproximately equal to the summass of the uniform thin sheet,the limit.xyo(,)x y(,)x y01lim(,).niiiiM =(3)(4)Problem IntroductionFind the Mass(physical meaning).),(lim10iiniiM =The volume of the top cylinder.),(lim10iini
5、ifV =Definition(,),(1,2,)iiifin=1(,),niiiif=(,)f x yAndAnd(,)ii Assume that is a bounded function on the bounded closed region D.The closed area D is arbitrarily divided into n small closed area where represents the i-th small closed area,pick a sample point on each 12,niilim0=1(,)Definitiondomain D
6、dyxf),(，N Namely,amely,Ddyxf),(iiniif =),(lim10.sumintegrandvariableexpressionArea element(1)In the definition of double integral,the division of the closed regionis arbitrary .(2)When f(x,y)is continuous in the closed area,the limit of the sum inthe definition must exist,that is,the double integral
7、 must exist.Some instructions Some instructions In the rectangular coordinate system,thearea D is divided by a straight line networkparallel to the coordinate axes.(,)(,)DDf x y df x y dxdy=ddxdy=So the double integral can be written asxyoDThe area element is Geometric Meaning(1)Recall that if()0，re
8、presents the area of theregion below the curve =()with from a to b.In a similar manner(2)if ,0，(,)represents the volume of thesolid below the surface z=f(x,y)and above the region D.()baf x dxIntegrability TheoremIf f is bounded on the closed region D and if it is continuous there except on a finite
9、number of smooth curves,then f is integrable on D.In particular,if f is continuous on all of D,then f is integrable there.Theorem AProperties(1)is a constant.k(,)(,).DDkf x y dkf x y d=(2)Ddyxgyxf),(),(,)(,).DDf x y dg x y d=(3).),(),(),(21 +=DDDdyxfdyxfdyxf (4)1.DDdd=)(21DDD+=Properties(5)If),(),(y
10、xgyxf(,)(,).DDf x y dg x y dSpecially,.),(),(DDdyxfdyxf then(6)（evaluation theorem）(,)，,，is the area of region D，then (,)Properties(7)（middle value theorem）If(,)is continuous on the closed region D,is the area ofregion D,then there is at least one point(,)on D that satisfies:,=(,)Example 1Estimate t
11、he value ofwhere D is the ellipse region:22()xyDIed+=22221xyab+=,12220ayxeee =+,222)(aDyxede +222().xyaDabedab e+The area of D is ,ab=As to D 2220ayx+,Example 2Estimate the value ofWhere D:0 x1，0y222216DdIxyxy=+,16)(1),(2+=yxyxf1(0)4Mxy=5143122=+=m(1,2)xy=So 4252 I The maximum value of on D is(,)f x yThe area2,=The minimum value isSummaryThe definition of double integralIntegrability theoremProperties of double integralDouble Integrals over Rectangles