Chapter 7 Section 5 Doule Integrals over Rectangular Regions 569 5 Doule Integrals over Rectangular Regions In Prolems 5 through 53, use the method of Lagrange multipliers to find the indicated maximum or minimum. You will need to use the graphing utility or the solve application on your calculator. 5. Maximize f(x, y) e xy x ln suject to x y 4. 5. Minimize f(x, y) ln (x y) suject to xy y 5. 5. Minimize f(x, y) suject to x y 7. x 3 xy y 53. Maximize f(x, y) suject to x y. Similarly, to evaluate xe x y y x In Chapters 5 and 6, you integrated a function of one variale f(x) y reversing the process of differentiation, and a similar procedure can e used to integrate a function of two variales f(x, y). However, since two variales are involved, we shall integrate f(x, y) y holding one variale fixed and integrating with respect to the other. For instance, to evaluate the partial integral xy dx you would integrate with respect to x, using the fundamental theorem of calculus with y held constant: xy dx x x y x () y () y 3 y xy dy, you integrate with respect to y, holding x constant: xy dy x 3 y y3 y x 3 ()3 x 3 ()3 3 x In general, partially integrating a function f(x, y) with respect to x results in a function of y alone, which can then e integrated as a function of a single variale, thus producing what we call an iterated integral f(x, y) dy dx Similarly, the iterated integral is otained y first integrating with respect to y, holding x constant, and then with respect to x. For instance, f(x, y) dx dy.
57 Chapter 7 Calculus of Several Variales or xy dx 3 dy y dy y3y y xy dy dx 3 x dx xx 3 x In our example, the two iterated integrals turned out to have the same value, and it can e shown that this will e true as long as the limits of integration are constants. We use this fact to define the doule integral over a rectangular region as follows: The Doule Integral over a Rectangular Region The doule integral R f(x, y) da over the rectangular region y R: a x, c y d d c a x is given y the common value of the two iterated integrals f(x, y) dy and c dx that is, f(x, y) da R a d d c d a c f(x, y) dx a dy; f(x, y) dy dx c d f(x, y) dx a dy Here is another example. EXAMPLE 5. Evaluate the doule integral xe R y da
Chapter 7 Section 5 Doule Integrals over Rectangular Regions 57 where R is the rectangular region x, y 5 using: (a) x integration first () y integration first Solution (a) With x integration first: 5 xe y da R () Integrating with respect to y first, you get 5 xe y da xe y dy dx R 5 5 3 )y5 (ey 3 (e5 e ) 3 (e5 ) xe y dx dy yx x e dy x 5 ey [() () ] dy 3 ey dy y x(e y )y5 y (e 5 ) x x dx [x(e 5 e )] dx x (e5 )[() () ] 3 (e5 ) CHOOSING THE ORDER OF INTEGRATION In the examples you have seen so far, the order of integration makes little if any difference. Not only do the computations yield the same result, ut the integrations are essentially of the same level of difficulty. However, as the following example illustrates, sometimes the order does matter. EXAMPLE 5. Evaluate the doule integral xe R xy da where R is the rectangular region x, y.
57 Chapter 7 Calculus of Several Variales Solution If you evaluate the integral in the order it will e necessary to use integration y parts for the inner integration: Then the outer integration ecomes Now what? Any ideas? On the other hand, if you use y integration first, oth computations are easy: g(x) e xy G(x) y exy xe xy dx x exyx y x x y y exyx xe xy dy dx xe xy dx dy x y y ey y dy xe xy x f(x) x f(x) y exy dx y y ey y y y dx [e x ] dx e x xx x (e ) e e 3 APPLICATIONS OF DOUBLE INTEGRALS Recall from Section of Chapter 6 that the definite integral f(x) dx of a nonnegative function f(x) of one variale gives the area of the region under the graph of y f(x) and aove the interval a x. A analogous argument for nonnegative functions of two variales yields the following formula for volume. a Volume Formula If f(x, y) for all points (x, y) in the rectangular region R, then the solid under the graph of f and aove the region R has volume given y V f(x, y) da R
Chapter 7 Section 5 Doule Integrals over Rectangular Regions 573 EXAMPLE 5.3 Find the volume under the graph of f(x, y) R: x, y. and aove the rectangular region Solution Since f(x, y) for all (x, y) in R, the volume is given y the doule integral x y da x y dx dy V R y dy y x x x dy ln yy y ln.347 cuic units x y y [() () ] dy ln ln AVERAGE VALUE In Section 3 of Chapter 6, you saw that the average value of a function f(x) over an interval a x is given y the integral formula AV a That is, to find the average value of a function of one variale over an interval, you integrate the function over the interval and divide y the length of the interval. The two-variale procedure is similar. In particular, to find the average value of a function of two variales f(x, y) over a rectangular region R, you integrate the function over R and divide y the area of R. a f(x) dx Average Value Formula The average value of the function f(x, y) over the rectangular region R is given y the formula AV f(x, y) da area of RR
574 Chapter 7 Calculus of Several Variales EXAMPLE 5.4 In a certain factory, output is given y the Co-Douglas production function Q(K, L) 5K 3/5 L /5 where K is the capital investment in units of $, and L is the size of the laor force measured in worker-hours. Suppose that monthly capital investment varies etween $, and $,, while monthly use of laor varies etween,8 and 3, worker-hours. Find the average monthly output for the factory. Solution It is reasonale to estimate the monthly output y the average value of Q(K, L) over the rectangular region R: K,,8 L 3,. The region has area A area of R ( ) (3,,8) 8 so the average output is AV 5K 3/5 L /5 da 8R 3, 8,8 3, 8,8 8 (5) 5 3, 8 8 (5) 5 8 (8/5 8/5 ) 5 7 L3, L7/5 L,8 8 (5) 5 8 5 7 (8/5 8/5 )[(3,) 7/5 (,8) 7/5 ] 5,8.3 5L /5 5 8 K8/5,8 5K 3/5 L /5 dk dl K K L /5 8/5 8/5 dl Thus, the average monthly output is approximately 5,8 units. dl
Chapter 7 Section 5 Doule Integrals over Rectangular Regions 575 JOINT PROBABILITY DENSITY FUNCTIONS y ; yr x ; y FIGURE 7.5 Square consisting of all points (x, y) for which x and. One of the most important applications of integration in the social, managerial, and life sciences is the computation of proailities. The technique of integrating proaility density functions to find proailities was introduced in Chapter 6, Section 4. Recall that a proaility density function for a random variale X is a nonnegative function f(x) such that the proaility that X is etween a and is given y the formula P(a X ) f(x) dx a In situations involving two random variales X and Y, you compute proailities y evaluating doule integrals of a two-variale density function. In particular, you integrate a joint proaility density function, which is a nonnegative function f(x, y) such that the proaility that X is etween a and and Y is etween c and d is given y the formula d P(a X and c Y d) The techniques for constructing joint proaility density functions from experimental data are eyond the scope of this ook and are discussed in most proaility and statistics texts. The use of doule integrals to compute proailities once the appropriate density functions are known is illustrated in the next example. EXAMPLE 5.5 Smoke detectors manufactured y a certain firm contain two independent circuits, one manufactured at the firm s California plant and the other at the firm s plant in Ohio. Reliaility studies suggest that if X measures the life span (in years) of a randomly selected circuit from the California plant and Y the life span (in years) of a randomly selected circuit from the Ohio plant, then the joint proaility density function for X and Y is f(x, y) e x e y c a d a c if x and y otherwise f(x, y) dx dy f(x, y) dy dx If the smoke detector will operate as long as either of its circuits is operating, find the proaility that a randomly selected smoke detector will fail within year. Solution Since the smoke detector will operate as long as either of its circuits is operating, it will fail within year if and only if oth of its circuits fail within year. The desired proaility is thus the proaility that oth X and Y. The points
576 Chapter 7 Calculus of Several Variales (x, y) for which oth these inequalities hold form the square R shown in Figure 7.5. The corresponding proaility is the doule integral of the density function f over this region R. That is, P( X and Y ).3996 e x e y dx dy (e )e y dy ex e yx x dy yy (e )e (e ) Thus, it is approximately 4% likely that a randomly selected smoke detector will fail within year. y P. R. O. B. L. E. M. S 7.5 P. R. O. B. L. E. M. S 7.5 Evaluate the following doule integrals. ln 3 4 3. x ydxdy. x ydydx 3. xe y dx dy 4. (x y) dy dx xy 5. 6. x e xy dy dx x dx dy 7. x ydydx 8. x y 9. dy dx. xy 3 5 3 y y dx dy y x x y dy dx In Prolems through 6, find the volume of the solid ounded aove y the graph of the function f(x, y) and elow y the given rectangular region R.. f(x, y) 6 x y; R has vertices (, ), (, ), (, ), (, ). f(x, y) 9 x y ; R has vertices (, ), (, ), (, ), (, )
Chapter 7 Section 5 Doule Integrals over Rectangular Regions 577 3. f(x, y) ; R: x, y 3 xy 4. f(x, y) e xy ; R: x, y ln 5. f(x, y) xe y ; R: x, y 6. f(x, y) ( x)(4 y); R: x, y 4 In Prolems 7 through, find the average value of the function f(x, y) over the given rectangular region R. 7. f(x, y) xy(x y); R has vertices (, ), (3, ), (, ), (3, ) y 8. f(x, y) ; R has vertices (, ), (4, ), (, 3), (4, 3) x x y 9. f(x, y) xe x y ; R: x, y ln x. f(x, y) ; R: x, y 3 xy In Prolems and, evaluate the given doule integral over the specified rectangular region R. Choose the order of integration carefully. ln xy. da over R: x 3, y 5 xy R. da over R: x, y R ye xy 3. Suppose the joint proaility density function for the nonnegative random variales X and Y is f(x, y) e x e y if x and y otherwise Find the proaility that X and Y. 4. Suppose the joint proaility density function for the nonnegative random variales X and Y is f(x, y) 6 ex/ e y/3 if x, y otherwise Find the proaility that X and Y. 5. Evaluate the doule integral x e xy da, where R is the rectangular region R x, y. (Be careful how you choose the order of integration.)
578 Chapter 7 Calculus of Several Variales PRODUCTION PRODUCTION AVERAGE PROFIT AVERAGE RESPONSE TO STIMULI AVERAGE ELEVATION AVERAGE SURFACE AREA OF THE HUMAN BODY da 6. Evaluate the doule integral, where R is the rectangular region R xy ln x x 3, y. 7. At a certain factory, output Q is related to inputs x and y y the expression Q(x, y) x 3 3x y y 3. If x 5 and y 7, what is the average output of the factory? 8. A icycle dealer has found that if -speed icycles are sold for x dollars apiece and the price of gasoline is y cents per gallon, then approximately Q(x, y) 4x 4(.y 5) 3/ icycles will e sold each month. If the price of icycles varies etween $57 and $3 during a typical month, and the price of gasoline varies etween $.3 and $., approximately how many icycles will e sold each month? 9. A manufacturer estimates that when x units of a particular commodity are sold domestically and y units are sold to foreign markets, the profit is given y P(x, y) (x 3)(7 5x 4y) (y 4)(8 6x 7y) hundred dollars. If monthly domestic sales vary etween and 5 units and foreign sales, etween 7 and 89 units, what is the average monthly profit? 3. In a psychological experiment, x units of stimulus A and y units of stimulus B are applied to a suject, whose performance on a certain task is then measured y the function P(x, y) xye x y Suppose x varies etween and while y varies etween and 3. What is the suject s average response to the stimuli? 3. A map of a small regional park is a rectangular grid, ounded y the lines x, x 4, y, and y 3, where units are in miles. It is found that the elevation aove sea level at each point (x, y) in the park is given y E(x, y) 9(x y ) feet Find the average elevation in the park. (Rememer, mi 5,8 feet.) 3. Recall from Prolem 35, Section, that the surface area of a person s ody may e estimated y the empirical formula S(W, H).7 W.45 H.75 where W is the person s weight in kilograms, H is the person s height in centimeters, and the surface area S is measured in square meters. (a) A child weighs 3. kg and is 38 cm tall at irth, and 5 years later weighs 4 kg and is 5 cm tall. What is the average surface area of the child s ody during this period? () Suppose the child in part (a) has a stale adult weight of 8 kg and height of 8 cm. Find the average lifetime surface area of this person s ody.
Chapter 7 Section 5 Doule Integrals over Rectangular Regions 579 HEALTH CARE WARRANTY PROTECTION PROPERTY VALUE EXPOSURE TO DISEASE 33. Suppose the random variales X and Y measure the length of time (in days) that a patient stays in the hospital after adominal and orthopedic surgery, respectively. On Monday, the patient in ed 7A undergoes an emergency appendectomy while her roommate in ed 7B undergoes (orthopedic) surgery for the repair of torn knee cartilage. If the joint proaility density function for X and Y is f(x, y) ex/4 e y/3 what is the proaility that oth patients will e discharged from the hospital within 3 days? 34. A certain appliance consisting of two independent electronic components will e usale as long as at least one of its components is still operating. The appliance carries a warranty from the manufacturer guaranteeing replacement if the appliance ecomes unusale within year of the data of purchase. Let the random variale X measure the life span (in years) of the first component and Y measure the life span (also in years) of the second component, and suppose that the joint proaility density function for X and Y is f(x, y) 4 ex/ e y/ if x and y otherwise if x and y otherwise You purchase one of these appliances, selected at random from the manufacturer s stock. Find the proaility that the warranty will expire efore your appliance ecomes unusale. [Hint: You want the proaility that the appliance does not fail during the first year. How is this related to the proaility that the appliance does fail during this period?] 35. Suppose a particular congressional district is laid out in a rectangular grid with a city at the origin. It is estimated that the property value at the point (x, y) in the grid is given y V(x, y) 9 e x y thousand dollars, where x and 5 y 5. Find the average property value for the district. [Hint: You will not e ale to find an antiderivative for e x. Estimate the average value integral using the numeric integration feature of your calculator.] 36. The likelihood that a person with a contagious disease will infect others in a social situation may e assumed to e a function f(s) of the distance s etween individuals. Suppose contagious individuals are uniformly distriuted through a rectangular region R in the xy plane. Then the likelihood of infection for someone at the origin (, ) is proportional to the exposure index E, given y the doule integral E f(s) da R where s x y is the distance etween (, ) and (x, y),
58 Chapter 7 Calculus of Several Variales (a) Find E for the case where R is the square region x, y, and f(s). 9 s () Find E for the case where R is the region x, 3 y, and f(s) se 5. (c) Find E for the case where R is the region in part () ut f(s) e.5. [Hint: Use the numeric integration feature of your calculator.] In Prolems 37 through 39, use doule integration to find the required quantity. In some cases, you may need to use the numeric integration feature of your calculator. 37. Find the volume of the solid ounded aove y the graph of f(x, y) x e xy and elow y the rectangular region R: x, y 3. 38. Find the average value of the function f(x, y) xy ln y over the rectangular region ounded y the lines x, x, y, and y 3. 39. Suppose the joint proaility density function for the random variales X and Y is f(x, y) xe xy if x and y otherwise Find the proaility that (X, Y ) lies in the rectangular region R with vertices (,), (,), (,), and (,). x CHAPTER SUMMARY AND REVIEW PROBLEMS CHAPTER SUMMARY AND REVIEW PROBLEMS IMPORTANT TERMS, SYMBOLS, AND FORMULAS Function of two variales: z f(x, y) Surface Level curve: f(x, y) C Constant-production curve (isoquant) Indifference curve Partial derivatives of z f(x,y): Second-order partial derivatives: f x z x f y z y