Administrative Announcements Midterm on Tuesday 3 18 6 45 8PM Conflict Wednesday 3 19 6 45 8AM No class on 3 19 3 21 Homework assignment due on 3 17 1 15 Multiple Integrals Copyright Cengage Learning All rights reserved 15 2 Iterated Integrals Copyright Cengage Learning All rights reserved 3 Iterated Integrals Suppose that f is a function of two variables that is integrable on the rectangle R a b c d We use the notation to mean that x is held fixed and f x y is integrated with respect to y from y c to y d This procedure is called partial integration with respect to y Notice its similarity to partial differentiation Now is a number that depends on the value of x so it defines a function of x 4 Iterated Integrals If we now integrate the function A with respect to x from x a to x b we get The integral on the right side of Equation 1 is called an iterated integral Usually the brackets are omitted Thus means that we first integrate with respect to y from c to d and then with respect to x from a to b 5 Iterated Integrals Similarly the iterated integral means that we first integrate with respect to x holding y fixed from x a to x b and then we integrate the resulting function of y with respect to y from y c to y d Notice that in both Equations 2 and 3 we work from the inside out 6 Example 1 Evaluate the iterated integrals a b Solution a Regarding x as a constant we obtain 7 Example 1 Solution cont d Thus the function A in the preceding discussion is given by in this example We now integrate this function of x from 0 to 3 8 Example 1 Solution cont d b Here we first integrate with respect to x 9 Iterated Integrals Notice that in Example 1 we obtained the same answer whether we integrated with respect to y or x first In general it turns out see Theorem 4 that the two iterated integrals in Equations 2 and 3 are always equal that is the order of integration does not matter This is similar to Clairaut s Theorem on the equality of the mixed partial derivatives 10 Iterated Integrals The following theorem gives a practical method for evaluating a double integral by expressing it as an iterated integral in either order 11 Iterated Integrals In the special case where f x y can be factored as the product of a function of x only and a function of y only the double integral of f can be written in a particularly simple form To be specific suppose that f x y g x h y and R a b c d Then Fubini s Theorem gives 12 Iterated Integrals In the inner integral y is a constant so h y is a constant and we can write since is a constant Therefore in this case the double integral of f can be written as the product of two single integrals 13 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 15 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 16 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 17 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 18 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 19 Double Integrals over General Regions Using the same methods that were used in establishing we can show that 20 Properties of Double Integrals 21 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 22 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 23 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 24 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 25 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 26 Properties of Double Integrals Finally we can combine Properties 7 8 and 10 to prove the following property 27
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