15 Multiple Integrals Copyright Cengage Learning All rights reserved 15 3 Double Integrals over General Regions Copyright Cengage Learning All rights reserved Double Integrals over General Regions Learning objectives evaluate double integrals over regions of type I and or type II as iterated integrals evaluate double integrals over regions that are the union of regions of type I and or type II find the region of integration given by the intersection of two or more plane curves find the area of plane regions find the volume of solids above a plane region and below the graph of a function of two variables reverse the order of integration of iterated integrals 3 Double Integrals over General Regions For single integrals the region over which we integrate is always an interval But for double integrals we want to be able to integrate a function f not just over rectangles but also over regions D of more general shape such as the one illustrated in Figure 1 Figure 1 4 Double Integrals over General Regions A plane region D is said to be of type I if it lies between the graphs of two continuous functions of x that is D x y a x b g1 x y g2 x where g1 and g2 are continuous on a b Some examples of type I regions are shown in Figure 5 Figure 5 Some type I regions 5 Double Integrals over General Regions The following formula enables us to evaluate the double integral over a type I region as an iterated integral The integral on the right side of is an iterated integral except that in the inner integral we regard x as being constant not only in f x y but also in the limits of integration g1 x and g2 x 6 Double Integrals over General Regions We also consider plane regions of type II which can be expressed as D x y c y d h1 y x h2 y where h1 and h2 are continuous Two such regions are illustrated in Figure 7 Figure 7 Some type II regions 7 Double Integrals over General Regions Using the same methods that were used in establishing we can show that 8 Properties of Double Integrals 9 Properties of Double Integrals We assume that all of the following integrals exist The first three properties of double integrals over a region D follow immediately from Definition 2 If f x y g x y for all x y in D then 10 Properties of Double Integrals The next property of double integrals is similar to the property of single integrals given by the equation If D D1 U D2 where D1 and D2 don t overlap except perhaps on their boundaries see Figure 17 then Figure 17 11 Properties of Double Integrals Property 9 can be used to evaluate double integrals over regions D that are neither type I nor type II but can be expressed as a union of regions of type I or type II Figure 18 illustrates this procedure D is neither type I nor type II Figure 18 a D D1 D2 D1 is type I D2 is type II Figure 18 b 12 Properties of Double Integrals The next property of integrals says that if we integrate the constant function f x y 1 over a region D we get the area of D 13 Properties of Double Integrals Figure 19 illustrates why Equation 10 is true A solid cylinder whose base is D and whose height is 1 has volume A D 1 A D but we know that we can also write its volume as D 1 dA Figure 19 Cylinder with base D and height 1 14 Properties of Double Integrals Finally we can combine Properties 7 8 and 10 to prove the following property 15
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