Multiple IntegrationCopyright Cengage Learning. All rights reserved. Double Integrals and VolumeCopyright Cengage Learning. All rights reserved. 3Use a double integral to represent the volume of a solid region and use properties of double integrals.valuate a double integral as an iterated integral.!ind the average value of a fun"tion over a region.#b$e"tives%Double Integrals and Volume of a &olid 'egion(Double Integrals and Volume of a &olid 'egion)ou *no+ that a definite integral over an interval uses a limit pro"ess to assign measures to ,uantities su"h as area- volume- ar" length- and mass.In this se"tion- you +ill use a similar pro"ess to define the double integral of a fun"tion of t+o variables over a region in the plane..Consider a "ontinuous fun"tion f su"h that f/x- y0 1 2 for all /x- y0 in a region R in the xy3plane. 4he goal is to find the volume of the solid region lying bet+een the surfa"e given byz 5 f/x- y0 &urfa"e lying above the xy3planeand the xy3plane- as sho+n in !igure 6%.7.Figure 14.8Double Integrals and Volume of a &olid 'egion8)ou "an begin by superimposing a re"tangular grid over the region- as sho+n in !igure 6%.9.4he re"tangles lying entirely +ithin R form an inner partition :- +hose norm ;;:;; is defined as the length of the longest diagonal of the n re"tangles.Figure 14.9Double Integrals and Volume of a &olid 'egion7e"ause the area of the ith re"tangle is:AiArea of ith re"tangleit follo+s that the volume of the ith prism isf/xi- yi0 :AiVolume of ith prismFigure 14.10Double Integrals and Volume of a &olid 'egion9and you "an appro=imate the volume of the solid region by the 'iemann sum of the volumes of all n prisms-as sho+n in !igure 6%.66.4his appro=imation "an be improvedby tightening the mesh of the grid toform smaller and smaller re"tangles-as sho+n in =ample 6.Figure 14.11Double Integrals and Volume of a &olid 'egion62=ample 6 ? Approximating the Volume of a SolidAppro=imate the volume of the solid lying bet+een the paraboloidand the s,uare region R given by 2 @ x @ 6- 2 @ y @ 6.

Use a partition made up of s,uares +hose sides have a length of 66=ample 6 ? Solution>egin by forming the spe"ified partition of R.!or this partition- it is "onvenient to "hoose the "enters of the subregions as the points at +hi"h to evaluate f/x- y0.6A=ample 6 ? Solution>e"ause the area of ea"h s,uare is you "an appro=imate the volume by the sum4his appro=imation is sho+ngraphi"ally in !igure 6%.6A.4he e=a"t volume of the solid is .Figure 14.12"ontBd63=ample 6 ? Solution)ou "an obtain a better appro=imation by using a finerpartition.!or e=ample- +ith a partition of s,uares +ith sides of length the appro=imation is 2...7."ontBd6%Double Integrals and Volume of a &olid 'egionIn =ample 6- note that by using finer partitions- you obtain better appro=imations of the volume. 4his observation suggests that you "ould obtain the e=a"t volume by ta*ing a limit. 4hat is-Volume6(Double Integrals and Volume of a &olid 'egion4he pre"ise meaning of this limit is that the limit is e,ual to L if for every C 2 there e=ists a C 2 su"h that for all partitions of the plane region R /that satisfy ;;;; D 0 and for all possible "hoi"es of xi and yi in the ith region.6.Double Integrals and Volume of a &olid 'egionUsing the limit of a 'iemann sum to define volume is a spe"ial "ase of using the limit to define a double integral. 4he general "ase- ho+ever- does not re,uire that the fun"tion be positive or "ontinuous.Eaving defined a double integral- you +ill see that a definite integral is o""asionally referred to as a single integral.68Double Integrals and Volume of a &olid 'egion&uffi"ient "onditions for the double integral of f on the region R to e=ist are that R "an be +ritten as a union of a finite number of nonoverlapping subregions/see !igure 6%.630 that are verti"ally or horiFontally simple and that f is "ontinuous on the region R.4his means that the interse"tionof t+o nonoverlapping regions is aset that has an area of 2. In the figure-the area of the line segment "ommonto R6 and RA is 2.Figure 14.1367A double integral "an be used to find the volume of a solid region that lies bet+een the xy3plane and the surfa"e given by z 5 f/x- y0.Double Integrals and Volume of a &olid 'egion69Groperties of Double IntegralsA2Groperties of Double IntegralsDouble integrals share many properties of single integrals.A6valuation of Double IntegralsAAvaluation of Double IntegralsConsider the solid region bounded by the plane

z 5 f/x- y0 5 A ? x ? Ay and the three "oordinate planes- as sho+n in !igure 6%.6%.Figure 14.14A3valuation of Double Integralsa"h verti"al "ross se"tion ta*en parallel to the yz3plane is a triangular region +hose base has a length of y 5 /A ? x0HA and +hose height is z 5 A ? x.4his implies that for a fi=ed value of x- the area of the triangular "ross se"tion isA%valuation of Double Integrals>y the formula for the volume of a solid +ith *no+n "ross se"tions- the volume of the solid is4his pro"edure +or*s no matter ho+ A/x0 is obtained. In parti"ular- you "an find A/x0 by integration- as sho+n in !igure 6%.6(.Figure 14.15A(valuation of Double Integrals4hat is- you "onsider x to be "onstant- and integrate z 5 A ? x ? Ay from 2 to /A ? x0HA to obtainCombining these results- you have the iterated integralA.valuation of Double Integrals4o understand this pro"edure better- it helps to imagine the integration as t+o s+eeping motions. !or the inner integration- a verti"al line s+eeps out the area of a "rossse"tion. !or the outer integration- the triangular "ross se"tion s+eeps out the volume- as sho+n in !igure 6%.6..Figure 14.16A8valuation of Double IntegralsA7=ample A ? Evaluating a Double Integral as an Iterated Integralvaluate+here R is the region given by 2 @ x @ 6- 2 @ y @ 6.&olutionI>e"ause the region R is a s,uare-it is both verti"ally and horiFontally simple-and you "an use either order of integration.Choose dy dx by pla"ing a verti"al representative re"tangle in the region-as sho+n in the figure at the right.A9=ample A ? Solution4his produ"es the follo+ing."ontBd32Average Value of a !un"tion36Average Value of a !un"tion!or a fun"tion fin one variable- the average value of f on

Ja- bK isLiven a fun"tion f in t+o variables- you "an find the average value of f over the region R as sho+n in the follo+ing definition.3A=ample . ? Finding the Average Value of a Funtion!ind the average value of f/x- y0 5over the region R- +here R is a re"tangle +ith verti"es /2- 20- /%- 20- /%- 30-and /2- 30.&olutionI4he area of the re"tangular regionR is A 5 6A /see !igure 6%.AA0.Figure 14.2233=ample . ? Solution4he average value is given by"ontBd