1、Section 10.11Riemann,Bernhard2Mass of a Thin Rectangular Sheet MetalSuppose a thin rectangular sheet metal lieson the xOy-plane and its density is a functionthen,how(,),(,),x yf x y of the point can we find the mass of this sheet metal?To find the mass,we suppose that(,)f x yis defined on a rectangu
2、lar region given by:,.axbcydWe divide into small pieces of areaand number them in some A order 12,.nAAAkA 3Mass of a Thin Rectangular Sheet Metalin each piece(,)kkxyWe choose a point kA and form the sum1(,).nnkkkkSf xyA If f is continuous throughout,then,aswe refine the mesh(or two-dimensional piece
3、 go to zero,partition)width to make the“norm”of eachsum should have a limit and the limit shouldbe the mass of the thin rectangular steelmetal.(,)kkxykA we can expect that the01lim(,).nkkkdkSf xyA 4Volume of a Cylindrical bodycylindrical body in threedimensional space.How can we find thevolume of th
4、is body?then it can be think of aSuppose that(,)0,(,)(),zf x yx y 5Volume of a Cylindrical bodycylindrical body in threedimensional space.How can we findthe volume of this body?then it can be think of aSuppose that(,)0,(,)(),zf x yx y 6Volume of a Cylindrical bodycylindrical body in threedimensional
5、 space.How can we find thevolume of this body?then it can be think of aSuppose that(,)0,(,)(),zf x yx y 7Volume of a Cylindrical bodycylindrical body in threedimensional space.How can we find thevolume of this body?then it can be think of aSuppose that(,)0,(,)(),zf x yx y 8Volume of a Cylindrical bo
6、dycylindrical body in threedimensional space.How can we find the volume of this body?then it can be think of aSuppose that(,)0,(,)(),zf x yx y 9Volume of a Cylindrical bodycylindrical body in threedimensional space.How can we find the volume of this body?then it can be think of aSuppose that(,)0,(,)
7、(),zf x yx y 10The Concept of the Double IntegralDefinition Double IntegralSuppose that a scalar function f is defined on a closed bounded is k pieces of area ,1,2,kkn and the measurement of and form the sum.k denoted by kkP Choose any point1().nkkkf P 01lim()nkkdkf P exist,where If the limit 1max()
8、,nkkdd we say that f is integrable().over the domainplane regionSuppose,however,can be divided into pieces of().()PartitionSummationPrecisionRiemann,Bernhard(1826-1866),German mathematician Page 222/definition 11.1.111The Notation of the Double Integralthen the limit of Riemann().If function f is in
9、tegral over the domain sum is called the integral of the multivariable function f on the domain(),01()(,)lim()nkkdkf x y df P Domain of integrationIntegrandIntegrand representationIntegral element Element of area12Properties of Double Integralthen(),Suppose(,)f x yand(,)g x yare both integrable over
10、 the domain 2.Additivity with respect to the domain of integration11()()()(,)(,)(,).f x y df x y df x y d,where k is a constant.()()(,)(,)kf x y dkf x y d(1)1.Linearity Property2()1()and12(),()Suppose that12()()()have no common part except for their Thenboundaries.(2)()()()(,)(,)(,)(,)f x yg x ydf x
11、 y dg x y d13Properties of Double Integral4.Mean Value Theoremon().then()()(,)(,)f x y dg x y d,if(,)(,)f x yg x y on().(1)()(,)0f x y d ,if(,)0f x y (2)(,),(,)(),lf x yLx y(4)If()(,).lf x y dL 3.Domination,such thatis a closed bounded,and connected()Suppose that ,()anddomain.()fC Then there exists at least one point ()(,),.f x y df ()()(,)(,)f x y df x y d(3)
