806 Chapter Fifteen OPTIMIZATION: LOCAL AND GLOBAL EXTREMA If f is not continuous or the region R is not closed and bounded, there is no guarantee that f achieves a global maximum or global minimum on R. In Example 4, the function g is continuous but does not achieve a global maximum or minimum in 2-space, a region which is closed but not bounded. Example 6 illustrates what can go wrong when the region is bounded but not closed. Example& Does the function f have a global maximum or minimum on the region R given by 0 < x 2 f(x, y) Solution = + y2 ~ 1? 1 x2 + y2 The region R is bounded, but it is not closed since it does not contain the boundary point (0, 0). We see from the graph of z = f(x, y) in Figure 15.25 that f has a global minimum on the circle x 2 + y 2 = 1. However, f(x, y) __. oo as (x, y) __. (0, 0), so f has no global maximum. z Figum 'i5.25: Graph showing f(x, y) = x 2 !Y2 has no global maximum on 0 < x 2 + y2 ~ 1 Exercises and Problems for Section 15.2 Exerciser. 1. By looking at the weather map in Figure 12.1 on page 638, find the maximum and minimum daily high temperatures in the states of Mississippi, Alabama, Pennsylvania, New York, California, Arizona, and Massachusetts. In Exercises 2-4, estimate the position and approximate value of the global maxima and minima on the region shown. 2. y 3. y y 4. 57r/2 27r 37r/2 7r ....... L X Tr/2 7 7r 15.2 OPTIMIZAnON In Exercises 5-7, find the global maximum and minimum of the function on -1 ~ x ~ 1, -1 ~ y ~ 1, and say whether it occurs on the boundary of the square. [Hint: Use graphs.] 807 Do the functions in Exercises 9-13 have global maxima and minima? 9. f(x,y) = x2 - 2y 2 10. g(x, y) = x 2 y 2 8. Compute the regression line for the points ( -1, 2), {0, -1), {1, 1} using least squares. 11. h(x, y) = x3 + y 3 12. f(x, y) = -2x2 - 13. f(x, y) = x 2 /2 + 3y3 + 9y2 - 7y2 3x Problems 14. (a) Compute the critical points of f(x, y) = 2x2 3xy + 8y 2 + x - y and classify them. (b) By completing the square, plot the contour diagram of f and show that the local extremum found in part (a) is a global one. 15. Find the parabola of the form y = ax 2 + b which best fits the points (1, 0), (2, 2), {3, 4} by minimizing the sum of squares, S, given by S = (a + b) 2 + (4a + b - 2} 2 + (9a + b - 4) 2 23. What is the shortest distance from the surface xy + 3x + z 2 = 9 to the origin? 24. Two products are manufactured in quantities Q1 and Q2 and sold at prices of P and P2 respectively. The cost of producing them is given by c= 2q~ + 2q~ + 10. (a) Find the maximum profit that can be made, assum ing the prices are fixed. (b) Find the rate of change of that maximum profit as P 16. A missile has a guidance device which is sensitive to both temperature, tC, and humidity, h. The range in km over which the missile can be controlled is given by Range = 27,800 - 5e - 6ht - 3h 2 + 400t + 300h. What are the optimal atmospheric conditions for controlling the missile? 3 17. A closed rectangular box has volume 32cm . What are the lengths of the edges giving the minimum surface area? 18. An open rectangular box has volume 32 cm3 What are the lengths of the edges giving the minimum surface area? 19. A closed rectangular box with faces parallel to the coordinate planes has one bottom comer at the origin and the opposite top comer in the first octant on the plane 3x + 2y + z = 1. What is the maximum volume of such a box? 20. An international airline has a regulation that each passenger can carry a suitcase having the sum of its width, length and height less than or equal to 135 em. Find the dimensions of the suitcase of maximum volume that a passenger may carry under this regulation. 21. Design a rectangular milk carton box of width w, length l, and height h which holds 512 cm 3 of milk. The sides of the box cost 1 cent/cm 2 and the top and bottom cost 2 centfcm2. Find the dimensions of the box that minimize the total cost of materials used. 22. Find the point on the plane 3x+ 2y+ z = 1 that is closest to the origin by minimizing the square of the distance. 5 Adapted increases. 25. A company operates two plants which manufacture the same item and whose total cost functions are c. = 8.5 + 0.03q~ and c2 = 5.2 + 0.04q~' where Ql and q2 are the quantities produced by each plant. The total quantity demanded, q = q1 + q2, is related to the price, p, by p = 60 - 0.04q. How much should each plant produce in order to maximize the company's profit? 5 26. The quantity of a product demanded by consumers is a function of its price. The quantity of one product demanded may also depend on the price of other products. For example, the demand for tea is affected by the price of coffee; the demand for cars is affected by the price of gas. The quantities demanded, q1 and q2, of two products depend on their prices, P and p2, as follows Ql = 150- 2pl - Q2 = 200 - P2 P1 - 3p2. (a) What does the fact that the coefficients of PI and P2 are negative tell you? Give an example of two products that might be related this way. (b) If one manufacturer sells both products, how should the prices be set to generate the maximum possible revenue? What is that maximum possible revenue? from M. Rosser, Basic: Mathematics for &mwmists, p. 318 (New York: Routledge, 1993). 15.3 CONSTRAINED OPTIMIZATION: LAGRANGE MULTIPLIERS so we solve the system of equations we get from grad= 81 S 0: 8 = 20x-3f5ylf5zl/5- SOA = 0, ax o, 8 8y = lOx2f5y-415zii5 - = 10x2f5ylf5z-4/5- lOA= 0, ac az ac BA _ 12 A = = -(SOx+ 12y + lOz - 24,000) = 0. We simplify this system to give 1 A= -x-3f5ylf5zifs, 4 A= ~x2f5y-4f5zii5, 6 A = x2f5yi/5 z-4/5, SOx + 12y + lOz = 24,000. Eliminating z from the first two equations gives x = 0.3y. Eliminating x from the second and third equations gives z = 1.2y. Substituting for x and z into SOx+ 12y + lOz = 24,000 gives S0{0.3y) + 12y + 10(1.2y) = 24,000, soy = 500. Then x = 150 and z = 600, and /(150, 500, 600) = 4,622 units. The graph of the constraint, SOx+ 12y+ lOz = 24,000, is a plane. Since the inputs x, y, z must be nonnegative, the graph is a triangle in the first quadrant, with edges on the coordinate planes. On the boundary of the triangle, one (or more) of the variables x, y, z is zero, so the function f is zero. Thus production is maximized within the budget using x = 150, y = 500, and z = 600. Exercises and Problems for Section 15.3 Exercises In Exercises 1-17, use Lagrange multipliers to find the maximum and minimum values off subject to the given constraint, if such values exist. 1. f(x, y) 2. j (X, = x + y, Y) = X + y2 = 1 x2 x 2 + y2 = 10 + 3y + 2, 3. f(x,y)=(x-1} 2 +{y+2) 2 , 4. 5. f(x,y) = 3x- 2y, x 6. f(x,y) 7. = xy, /(XI, X2) 8. f(x, y) = 2 2 + 2y 11. f(x, y, z) = xyz, x = x + 2y f(x, y) = x + 3y, 2 14. f(x, y) = xy, x 2 2 2 x +y+z 2 2 x +y ~4 , x 2 2 +y + 2y 2 2 ~2 ~ 1 + y2 ::; 1 = (x + 3) + (y- 3} , x 2 + y 2 ~ 2 = x 2 y + 3y2 - y, x 2 + y2 ~ 10 2 18. Decide whether each point appears to be a maximum, minimum, or neither for the function f constrained by the loop in Figure 15.30. p (b) Q (c) R (d) I I I 10 Constraint x 2 +y 2 + z 2 = 1 2 x2 2 I= 2 + y + 4z = 12 2 17. f(x,y) y2, - = 44 2 10. f(x, y, z} = 2x + y + 4z, 12. f (x, y) 2 16. f(x, y) (a) +y =8 2 = X1 + X2 2 , XI + X2 = 1 x 2 + y, x 2 - y 2 = 1 4x 2 9. f(x,y,z) = x + 3y+ 5z, 13. x 2 +y 2 =5 f (x, y) = x + y, 3x + y = 4 2 3 15. f(x,y) = x 3 p = 16 . . . . . . _._-......._~ s / 1 =5o , 1_ = 30 1 =:= 40 j 1 =_ 6o Figure 15.30 s Chapter Fifteen OPTIMIZAnON: LOCAL AND GLOBAL EXTREMA 81 & Probiems 19. Find the maximum value of f(x, y) = x on the triangular region x ;:::: 0, y ;:::: 0, x 20. (a) Draw contours of f(x, y) = 2x + y- (x- y) 2 + y ::; 1. +y = for z -7, -5, -3, -1, 1, 3, 5, 7. (b) On the same axes, graph the constraint x 2 + y 2 = 5. (c) Use the graph to approximate the points at which f has a maximum or a minimum value subject to the constraint x 2 + y 2 = 5. (d) Use Lagrange multipliers to find the maximum and minimum values of f(x, y) = 2x + y subject to x2 + y2 = 5. and income, 8. If. for example, you are indifferent between 0 hours of leisure and an income of $1125 a week on the one hand, and 10 hours of leisure and an income of $750 a week on the other hand, then the points l = 0, 8 = 1125, and l = 10, 8 = 750 both lie on the same indifference curve. Table 15.3 gives information on three indifference curves, I, II, and Ill. Table Weekly income II 21. Let S be a surface with a wire rim, R, as its boundary and let g(x, y, z) be the temperature, in degrees Celsius, at a point (x, y, z). The hottest point on the wire rim R is P. At P, we are told that gradg = that the wind and that = 2. Decide if each of the velocity is statements (a)-(e) is w. v w C (meaning could be true, could be false). v w v is parallel to (a) (b) Angle () between and w satisfies 1r /2 < () ::; (c) At least one of llv II or llw II is greater than I (d) is tangent to the wire rim Rat P (e) is tangent to the wire rim R at P 1r v w 22. Figure 15.31 shows contours of f. Does f have a maximum value subject to the constraint g(x, y) = c for x ~ 0, y ~ 0? If so, approximately where is it and what is its value? Does f have a minimum value subject to the constraint? If so, approximately where and what? 16 14 Ill 750 500 I l II l Ill 1 40 50 60 30 ;l 50 i. ............ 50 1 70 1 90 10 20 Lao 1 40 1 ~ 375 250 70 (a) Graph the three indifference curves. (b) You have 100 hours a week available for work and leisure combined, and you earn $10/hour. Write an equation in terms of l and 8 which represents this constraint. (c) On the same axes, graph this constraint. (d) Estimate from the graph what combination of leisure hours and income you would choose under these circumstances. Give the corresponding number of hours per week you would work. 24. Figure 15.32 shows V f for a function f(x, y) and two curves g(x, y) = 1 and g(x, y) = 2. Mark the following: (a) (b) (c) (d) (e) y Weekly leisure hours ....u ........2.....5.....:.....................;.......................1..... ~.....i... 3.9.. . L...... v, T (meaning must be true), F (meaning must be false), 15.~i 12 The point(s) A where f has a local maximum. The point(s) B where f has a saddle point. The point C where f has a maximum on g = 1. The point D where f has a minimum on g = 1. If you used Lagrange multipliers to find C, what would the sign of A be? Why? 10 \\ 8 6 4 g=2 -...jj--~--=T~ ~ y""~ ~-"'-~ \\ ~ -f>-1+ j 2 2 4 6 8 10 12 14 16 Figure 15.31 23. Each person tries to balance his or her time between leisure and work. The tradeotT is that as. you work less your income falls. Therefore each person has indifference curves which connect the number of hours of leisure, l, g=1 I,.. + I ~\\~ \\ t rt -.. ~ ~ ~ / ~ h:"' ........*" ..,_ 1f , ~'-~-J"::/ / t '-..t_.t. /I t \\ t It t \\ t ; I ' t t t t Figure 15.a.2 ' t \\ 1 t t \\ \\'>' .. t f tt ' ' \\ \\ 832 Chapter Sixteen INTEGRATING FUNCTIONS OF SEVERAL VARIABLES 3. Figure 16.6 shows contours of g(x, y) on the region R, with 5 $ x ~ 11 and 4 ~ y $ 10. Using ~x = ~Y = 2, find an overestimate and an underestimate for 5. Let R be the rectangle with vertices (0, 0), (4, 0), (4, 4), and (0, 4) and let f(x, y) = .JXY. (a) Find reasonable upper and lower bounds for fng(x,y)dA. fn fdA without subdividing R. (b) Estimate fn fdA by partitioning R into four subrectangles and evaluating fat its maximum and minimum values on each subrectangle. 6. Table 16.6 gives values of z = f(x, y) on the rectangle RwithO $ x ~ 6and0 ~ y ~ 8. (a) Estimate fn f(x,y)dA as accurately as possible. (b) Estimate the average value of f(x, y) on R. Table 16.5 X 4. Figure 16.7 shows contours of f (x, y) on the rectangle R with 0 $ x ~ 30 and 0 $ y $ 15. Using ~x = 10 and ~Y = 5, find an overestimate and an underestimate for fn f(x, y)dA. 0 0.... y 8 .......... y ~~~~. ~. 2 ...... .. 4 ....... ~'\\.... 10 ~ 6 81 68 55 7. Using Riemann sums with four subdivisions in each di- ...... rection, find upper and lower bounds for the volume under the graph of f(x, y) = 2 + xy above the rectangle R with 0 ~ x $ 2, 0 ~ y ~ 4. 1-=----~6~~.~--+--"-',_,,~~ r--~~ 5 4 3 100 ........ 90 85 ,.. 79 65 61 '" ""', X 30 20 Figure 16.7 8. If f(x, y) gives pollution density, in micrograms per square meter, and x and y are in meters, give the units and a practical interpretation of fn f(x, y)dA. Probiems In Problems 9-24, decide (without calculation) whether the integrals are positive, negative, or zero. Let D be the region inside the unit circle centered at the origin, let R be the right half of D and let B be the bottom half of D. 9. J0 dA J0 5xdA 15. fa(y 3 + y 5 ) dA 17. fa(Y- y 3 ) dA 19. 21. 23. population density of flies and mosquitos at various points in a rectangular study region. The graphs of the two population densities for the region are shown in Figures 16.8 and 16.9. Assuming that the units along the corresponding axes are the same in the two graphs. are there more flies or more mosquitos in the region? 10. fa dA 11. fn5xdA 13. 25. A biologist studying insect populations measures the J0 sin ydA J0 ex dA J0 xy 2 dA 12. fa 5xdA 14. 16. J0 (y 3 + y 5 ) dA I R (y a + y s ) dA flies mosquitos 18. f 0 (y- y 3 ) dA 20. J0 cosydA 22. f 0 xex dA 24. J8 xcosydA latitude figure 16.8 latitude figure 16.9 840 Chapter Sixteen INTEGRATING FUNCTIONS OF SEVERAL VARIABLES from x = 0 to x = 3y and that there is a strip for every y from 0 to 2. Thus, when we change the order of integration we get 1. 1x/3 6 2 2 xJY3+} dydx 3 = /. x.Jy 3 /. Y 0 0 0 + 1 dxdy. Now we can at least do the inner integral because we know the antiderivative of x. What about the outer integral? /. 2!.3y x.Jy3 + 1 dxdy = /.2 (~2.Jy3 + 1) lx=3y dy = /.2 +(y g 2 3 + 1) 1 dy 1 2 0 0 0 x=O 0 = (y3 + 1) 3' { = 27- 1 = 26. Thus, reversing the order of integration made the integral in the previous problem much easier. Notice that to reverse the order it is essential first to sketch the region over which the integration is being performed. Exercises and Problems for Section 16.2 E~ercises In Exercises 1-4, sketch the region of integration. 1. !." !.' 3. [ [ 0 2. ysinxdydx 4. y2 o y 1 ",, I-"'"--- x-2 [[' 15. y r, [[~" ydydx 0 xydxdy 14. 2 6 [ X 3 ...., / I /\\~ \\ \\ \\ \\ ~~------------l y2xdxdy o I I 3 X ~ 5 For Exercises 5-11, evaluate the integral. 16. 5. 4 3 /. 6. (4x + 3y) dxdy /. f. f. 2 17. y 3 (x 2 + y 2 )dydx y 2 3 '~--] 2 /"" 3 2 7. /. 9. J.' J.' .,.r'.;-""' 8. [ 6xydydx /. 2 [ x ydydx /." sin x dx dy 2 1. 1.-./li 2xcos(x 2 /. ) 3 13. y 2 0 X 2 3 4 For Exercises 18-23, sketch the region of integration and evaluate the integral. dxdy For each of the shaded regions R in Exercises 12-17, write j~ fdA as an iterated integral. 12. 3 3 10. ye' dx dy 2 11. 0 X r------------1l '"~-,--"/ i I' l-----------J ' 0 ....______.__ __.__.______._ 2 3 4 I i 1 r I X { },1 I I II /I ~ 20. ex+y dydx !.2x sinxdydx x } 1 3 22. /. 1/ 1:....----x 4 4 /. {5 y 12 j 18. 2 /. ' -2 2 + y 2 ) dydx 0 0 23. /_ (x /_ -y'9-x2 2xy dy dx 16.21TERATED INTEGRALS For Exercises 24-28, evaluate the integral. 24. 26. In ..jx + y dA, where R is the rectangle 0 :5 x 0 ~ y ~ I R (5x2 + 1) sin 3y dA, where R is the rectangle -1 ::5 X :5 1, 2. 27. 84t :5 1, 0 ::5 y ::5 7t/3. I R xy dA, where R is the triangle x + y :5 1, x 2:: 0, y 2: 0. 25. Calculate the integral in Exercise 24 using the other order of integration. 28. IR(2x + 3y) 2 dA, where R is the triangle with vertices at ( -1, 0), (0, 1), and (1, 0). Problems 29. Find the volume under the graph of the function f(x, y) = 6x2 y over the region shown in Figure 16.21. 8 /l / 6 0 X 1 2 Figure 16.21 30. Find the average value of f(x, y) angle 0 ::5 x ::5 3 and 0 ::5 y ::5 6. ell _IX dxdy nx I:x+8 f(x,y)dydx + Io4 Io-2x+8 J(x,y)dydx 41. The region W lies below the surface f(x, y) = 2 2 2e- 0. xydA. 51. The region R bounded by the graph of ax+ by+ cz = 1 and the coordinate planes. Assume a. b, and c > 0. 52. Find the average distance to the x-axis for points in the region bounded by the x-axis and the graph of y = x-x 2 53. Show that for a right triangle the average distance from any point in the triangle to one of the legs is one-third the length of the other leg. (The legs of a right triangle are the two sides that are not the hypotenuse.) 56. Give the contour diagram of a function f whose average value on the square 0 ~ x ~ 1. 0 ~ y ~ 1 is (a) Greater than the average of the values off at the four comers of the square. (b) Less than the average of the values off at the four comers of the square. 57. A rectangular plate of sides a and b is subjected to a normal force (that is, perpendicular to the plate). The pressure, p. at any point on the plate is proportional to the square of the distance of that point from one comer. Find the total force on the plate. [Note that pressure is force per unit area.] 16~3 TRIPLE INTEGRALS A continuous function of three variables can be integrated over a solid region W in 3-space in the same way as a function of two variables is integrated over a flat region in 2-space. Again, we start with a Riemann sum. First we subdivide W into smaller regions, then we multiply the volume of each region by a value of the function in that region, and then we add the results. For example, if W is the box a ::; x ::; b, c ::; y ::; d, p ::; z ::; q, then we subdivide each side into n, rn, and l pieces, thereby chopping W into nml smaller boxes, as shown in Figure 16.23. z q p Fi~1ure 16.23: Subdividing a three-dimensional box The volume of each smaller box is !:J..V = llxllyllz, 16.3 TRIPLE INTEGRALS 845 J There is a stack for every point in the xy-plane in the shadow of the cone. The cone z = x 2 + y2 2 2 intersects the horizontal plane z = 3 in the circle x + y = 9, so there is a stack for all (x, y) in the region x 2 + y 2 ~ 9. Lining up the stacks parallel to they-axis gives a slice from y = -v'9- x 2 to y = J9="X2, for each fixed value of x. Thus, the limits on the middle integral are 1 ~13 zdzdy. Jx2+y2 -v'9-x2 Finally, there is a slice for each x between -3 and 3, so the integral we want is Mass= 131~13J -3 - v'9-x2 zdzdydx. x2 +y2 Notice that setting up the limits on the two outer integrals was just like setting up the limits for a double integral over the region x 2 + y 2 ~ 9. As the previous example illustrates, for a region W contained between two surfaces, the innermost limits correspond to these surfaces. The middle and outer limits ensure that we integrate over the "shadow" of W in the xy-plane. Limits on Triple Integrals The limits for the outer integral are constants. The limits for the middle integral can involve only one variable (that in the outer integral). The limits for the inner integral can involve two variables (those on the two outer integrals). Exercises and Problems for Section 16.3 Exercises ------------------------------------------------------------------------------In Exercises 1-4, find the triple integrals of the function over 1y'l:::;21Jt-x2-:2 7. the region W. 1 1 0 1. f(x, y, z) = x 2 + 5y2 - z, W is the rectangular box 0 ~ X S 2, -1 ~ y ~ 1, 2 ~ Z ~ 3. 2. h(x, y, z) = ax+ by+ cz, W is the rectangular box 0 ~X~ 1, 0 ~ y ~ 1, 0 ~ Z ~ 2. 8. 9. 3. f(x, y, z) = e-;~;-y-:, W is the rectangular box with corners at (0, 0, 0), (a, 0, 0), (0, b, 0), and (0, 0, c). 4. f(x, y, z) = sinxcos(y+z), W is the cube 0 S x ~ 0 ~ y 'Tr, 0 ~ z ~ 'Tr. s Sketch the region of integration in Exercises 5-13. 5. 111 1y'l:::;2 1 0 6. -1 1111 1y'l:::;2 0 -1 f(x, y, z) dydzdx 0 0 f(x,y,z)dzdxdy 1r, f(x,y,z)dydxdz 0 11 1y'l:::;2 1 y'l:::;2 1111 1y'l:::;2 1 - 1 0 - o -1 -v1-x2 1 10. -Jt-:2 f(x,y,z)dydzdx f(x,y,z)dzdxdy 1y'l:::;21Jl-;~:2-:2 f(x, y, z) dydzdx 1- -vt-x2 o 1 1y'l:::;21Jl-y2-:2 11. f(x, y, z) dxdydz 1o -Jt-:2 -Jt-y2-:2 1 1y'l:::;21Jt-:r;2-:2 12. f(x,y,z)dydxdz 10 0 -Jt-:r;2-:2 1 1~ 1Jt-:r;2-y2 13. f(x,y,z)dzdxdy 1o o -v-z2-y2 846 Chapter Sixteen INTEGRATING FUNCTIONS OF SEVERAL VARIABLES Pmblems In Problems 14-19, write a triple integral, including limits of integration, that gives the specified volume. 14. Between z = x 0 ~ X ~ 11 0 +y ~ y ~ and z = 1 + 2x + 2y and above 2. 15. Between the paraboloid z = x 2 + y 2 and the sphere x 2 + y 2 + z 2 = 4 and above the disk x 2 + y 2 ~ 1. 16. Between 2x above x + y + 2y + z ~ 1, x ~ = 6 and 3x 0, y ~ 0. + 4y + z 17. Between the top portion of the sphere x 2 and the plane z = 2. = 6 and + y2 + z2 = 9 18. Under the sphere x 2 + y 2 + z 2 = 9 and above the region between y = x and y = 2x - 2 in the xy-plane in the first quadrant. 19. Under the sphere x + y + z = 4 and above the region x 2 + y 2 ~ 4, 0 ~ x ~ 1, 0 ~ y ~ 2 in the xy-plane. 2 2 2 In Problems 20-23, write limits of integration for the integral fw f(x, y, z) dV where W is the quarter or half sphere or cylinder shown. 20. 29. Find the volume of the region bounded by z = x 2 , 0 ~ x ~ 5, and the planes y = 0, y = 3, and z = 0. 30. Find the volume between the planes z = 1 + x + y and x + y + z = 1 and above the triangle x + y ~ 1, x ~ 0, y ~ 0 in the xy-plane. 31. Find the volume between the plane x + y + z = 1 and the xy-plane. for x + y $ 2, x ~ 0, y ~ 0. 32. Find the mass of a triangular-shaped solid bounded by the planes z = 1 + x, z = 1 - x, z = 0, and with 0 ~ y ~ 3. The density is o = 10 - z gm/(cm) 3 , and x, y, z are in em. 33. (a) What is the equation of the plane passing through the points {1, 0, 0). {0, 1, 0), and {0, 0, 1)? (b) Find the volume of the region bounded by this plane and the planes x = 0, y = 0, and z = 0. 34. A solid region Dis a half cylinder of radius I lying horizontally with its rectangular base in the xy-plane and its axis along they-axis from y = 0 toy= 10. (The region is above the xy-plane.) z 21. (a) What is the equation of the curved surface of this half cylinder? (b) Write the limits of integration of the integral f(x, y, z) dV in Cartesian coordinates. 0 Ir J y X X r ' r Y z 22. 23. rl xr 35. Find the mass of the solid bounded by the xy-plane, yzplane, xz-plane, and the plane (x/3) + (y/2) + {z/6) = 1, if the density of the solid is given by o(x, y, z) = x+y. 36. Find the mass of the pyramid with base in the plane z = -6 and sides formed by the three planes y = 0 andy- x = 4 and 2x + y + z = 4, if the density of the solid is given by o(x, y, z) = y. 37. Find the average value of the sum of the squares of three numbers x. y, z. where each number is between 0 and 2. J 24. Find the volume of the region bounded by z = x+y, z = 10, and the planes x = 0. y = 0. 25. Find the volume of the pyramid with base in the plane z = -6 and sides formed by the three planes y = 0 and y - x = 4 and 2x + y + z = 4. Let W be the solid cone bounded by z = x 2 + y 2 and z = 2. For Problems 38-46. decide (without calculating its value) whether the integral is positive, negative, or zero. 38. fw J x2 + y 2 dV 39. fw(z-Jx 2 40. fw x dV 41. fw ydV 26. Find the volume of the region bounded by the planes Z = 3y, Z = y, y = 1, x = 1 1 and X = 2. 42. j~ Z dV 43. fwxydV 27. Find the volume of the region between the plane z = x and the surface z = x 2 , and the planes y = 0, and y = 3. 44 28. Find the volume of the region bounded by z = x + y, 0 ~ x ~ 5, 0 ~ y ~ 5, and the planes x = 0, y = 0, and z = 0. J, w xyz dV 46. fw e-xyz dV + y 2 ) dV 45. fw(z- 2) dV 16.4 DOUBLE INTEGRALS IN POLAR COORDINATES Let W be the solid half-con\\! bounded by z = Jx 2 + y 2 z 2 and the y z -pl ane w ith x ~ 0. For Problems 47-55. decide (without calculating its val ue) whether the integral is positi ve, negative, or zero. = 47. j~ Jx2+y2dV 48. f w(z- Jx2 49. f wxdV so. 51. J~v z dV w e-xyz The motion of a sol id object can be analyzt!d by thinking of the mass as concentrated at a single point. the center of mass. If the object has density p(x , y , z) at the poi nt (x, y , z) and occupies a region W. then the coordi nates (x, y, z) o f the center of mass are given by - 11 x = - f w ydV 111 xpd\\1 W 52. f wxyd\\1 53. f wxyzd\\1 55. f + y2) d\\1 54. f w(z - 2) dV dV 847 - 11 y = - 111 ypd\\1 I ; zp dV z- = 1n \\V w J;,, where m = p d\\1 i s the total mass o f the body. Use these definitions for Problems 59-60. 59. A solid is bounded bel ow by the square z 1, 0 ~ y ~ 1 and above by the surface z = 0, 0 ~ x ~ = x + y + l. Find the tota l mass and the coordinates o f the center o f 3 mass if the density is 1 grn/cm and x , y , z are measured in centimeters. 56. Figure 16.26 shows part o f a spherical ball of radius 5 em. Wri te an iterated tri ple integral w hich represents the volume of this region. 60. Fi nd the center o f mass of the tetrahedron that i s bounded by the xy, y z. xz planes and the pl ane x + 2y + 3z 1. Assume the density is I grn/cm3 and x, y, z are in centimeters. = Figure 16.26 57. Set up, but d o volume o f the 2 cyli nders x + not evaluate, an iterated integra l for the solid formed by the intersections of the 2 2 2 z = 1 and y + z = 1. 58. A cube o f side length 2m i s composed o f t wo materials, separated by a plane slanting dow n from the top corner to cut the bottom face in a diagonal , as in Figure 16.27. The material below the plane has density 6 1 gm/m 3 and the material above the plane has density 62 grn!m 3 . Fi nd the mass of the cube. The moment of inertia o f a solid body about an axi s in 3-space relates the angular acceleration about thi s axis to torque (force twisti ng the body). The moments o f inertia about the coordinate axes of a body o f constant density and mass m occupying a region W of volume V arc defined to be fx 1n = \\1 j (y2 w 1TL J + Z 2 ) d\\1 fy 1 = 171 11 (x 2 + z 2 ) d\\1 w (x 2 + y-? )dV lz = V IV Use these definitions for Problem s 6 1- 63. 61. Find the moment o f inertia about the z -axi s of the rectangular solid of mass m. given by 0 ~ x ~ l , 0 ~ y ~ 2, 0 ~ z ~ 3. 62. Fi nd the moment of inertia about the x-axis of the rectangular sol id -a ~ x ~ a. -b ~ y ~ band -c ~ z ~ c of mass m. Figure 16.27 16.4 63. L et a. b, and c denote the moments o f inertia of a homogeneous solid object about the x, y and z -axes respecti vely. Explain w hy a+ b > c. DOUBLE INTEGRALS IN POLAR COORDINATES Integration in Polar Coordinates We started this chapter by putting a rectangular grid on the fox population density map, to estimate the total population using a Riemann sum. However, sometimes a polar grid is more appropriate. Example 1 A biologist studying insect popu lations around a circular lake div ides the area into the polar sectors in Figure 16.28. The population density in each sector is shown in millions per square km. Estimate the total insect population around the lake. 850 Chapter Sixteen INTEGRATING FUNCTIONS OF SEVERAL VARIABLES Solution (a) Since this is a rectangular region, Cartesian coordinates are likely to be a better choice. The rectangle is described by the inequalities 1 ::; x ::; 3 and -1 ::; y ::; 2, so the integral is 2 3 { /_ f(x, y) dx dy. ~ -111 (b) A circle is best described in polar coordinates. The radius is 3, so r goes from 0 to 3, and to describe the whole circle, 8 goes from 0 to 21r. The integral is {21r {3 10 10 f(r cos 8, r sin 8) 1 dr d8. (c) The bottom boundary of this trapezoid is the line y = (x/2) - 1 and the top is the line y = 3, so we use Cartesian coordinates. If we integrate with respect to y first, the lower limit of the integral is (x/2) -1 and the upper limit is 3. The x limits are x = 0 to x = 2. So the integral is 2 3 { f(x, y) dy dx. { 1o 1cx/2)-1 (d) This is another polar region: it is a piece of a ring in which r goes from 1 to 2. Since it is in the second quadrant, 8 goes from 1r /2 to 1r. The integral is 2 7r /. 7r/2 { 1. J(r cos 8, r sin 8) r dr d8. Exercises and Problems for Section 16.4 Exercis{~S ~*-~~~~~~~~~@0~~,~~~~~~~~-- For the regions R in Exercises 1-4, write ated integral in polar coordinates. 2. y 1. /~--~-- ',, ( fn fdA as an iter- 5. "'',,"" . '\\ 4-------f-----i- X }12 -.fZ., ~ y 0 5 1.. -~-...,,, '-,\\ In Exercises 5-8, choose rectangular or polar coordinates to set up an iterated integral of an arbitrary function f(x, y) over the region. '~-"''/ ""'-v'2 //,, ....~ -~-,,," ' 5 ;}~"" .........,. I ". .,.,"',-, I I ' \\ ~ y 6. y 5 3 l! '"'-'"-. I / I c----//- /.r_,."~< x 2 3. 4. y L3 // 1 ' ; l -2\\ -1,._,, ~- 2y ,.---- 2 X 7. / "~' ... X -1 y 4 8. -4 -2 r----l y 2 4 -L---4---~--!--+ \\ \\ : i X ~~2-<; -4-- - 2 X 5 J 16.4 DOUBLE INTEGRALS IN POLAR COORDINATES Sketch the region of integration in Exercises 9-15. 9. 1 -7r/2 411r/2 f(r, 0) r d8 dr 1 16. 0) r dr dO 0 11. 10 1 1 13.1 1 rr/6 4 14. 1 JR(x 2- y 2 J: J: 13 f(x, y) dy dx. (a) Sketch the region Rover which the integration is be- f(r,O}rdOdr ing performed. 3rr/4 (b) Rewrite the integral with the order of integration re- rr/411/cos9 f(r,O}rdrdO 0 JR sin(x 2 + y 2 } dA, where R is the disk of radius 2 centered at the origin. 19. Consider the integral 12 1r 3 f(r,O)rdrdO 0 3 f(r, 0) r dr dO 0 JR 1 12.11r/ 17r/4 ) dA, where R is the first quadrant region between the circles of radius 1 and radius 2. 18. Jx 2 + y 2 dxdy where R is 4 :5 x 2 + y 2 :59. 17. 27r 12 f(r, 0) r dr dO 3 1r /212/ sin 9 In Exercises 16-18, evaluate the integral. 0 10.11r7r/2 1f(r, 15. 351 versed. (c) Rewrite the integral in polar coordinates. 0 Problems Convert the integrals in Problems 20-22 to polar coordinates and evaluate. 20. l,;t::;i 1-1 -V1-z2 22. 1 ../21~ 0 xdydx " 21. v'61z 1 0 -;r; dydx xydxdy Figure 16.33 0 23. Find the volume of the region between the graph of f(x, y) = 25- x 2 - y 2 and the xy plane. 24. Find the volume of an ice cream cone bounded by the hemisphere z = x 2 - 2 and the cone z = Jx2 +y2. Js- y 25. Find the average distance from the center of the points inside a circle of radius a. 26. A city surrounds a bay as shown in Figure 16.33. The population density of the city (in thousands of people per square km) is 6(r, 0), where rand 0 are polar coordinates and distances are in km. (a) Set up an iterated integral in polar coordinates giving the total population of the city. (b) The population density decreases the farther you live from the shoreline of the bay; it also decreases the farther you live from the ocean. Which of the following functions best describes this situation? (i) 6(r, 0) = (4- r)(2 +cosO) (ii) 6(r, 0) y(km) = (4- r)(2 +sin 0) (iii) 6(r, 0) = (r + 4)(2 +cosO) (c) Estimate the population using your answers to parts (a) and (b). 27. (a) For a > 0, find the volume under the graph of 2 2 z = e- above the disk x 2 + y 2 :5 a 2 (b) What happens to the volume as a--+ oo? 28. A disk of radius 5 em has density I0 gm/cm 2 at its center, density 0 at its edge, and its density is a linear function of the distance from the center. Find the mass of the disk. 29. A watch spring lies flat on the table. It is made of a coiled steel strip standing a height of 0.2 inches above the table. The inner edge is the spiral r = 0.25 + 0.040, where 0 :5 0 ~ 47r (so the spiral makes two complete turns). The outer edge is given by r = 0.26 + 0.040. Find the volume of the spring. 30. Electric charge is distributed over the xy-plane. with density inversely proportional to the distance from the origin. Show that the total charge inside a circle of radius R centered at the origin is proportional to R. What is the constant of proportionality? 31. (a) Sketch the region of integration of 11V4-z2 xdydx + 121V4-z2 xdydx 10 V1-z2 1 0 (b) Evaluate the quantity in part (a). !bZ Chapter Sixteen INTEGRATING FUNCnONS OF SEVERAL VARIABLES 32. (a) Sketch the circles r = 2 cos 8 for -7r /2 ~ 8 ~ 1r /2 and r = 1. (b) Write an iterated integral representing the area inside the circle r = 2 cos 8 and outside the circle r = 1. Evaluate the integral. are I unit apart. Write, but do not evaluate, a double integral, including limits of integration, giving the area of overlap of the disks in (a) Cartesian coordinates (b) Polar coordinates 35. A thin circular disk of radius 12 em has density which increases linearly from I gm/cm 2 at the center to 25 gm/cm2 at the rim. 33. (a) Graph r = 1J(2 cos 8) for -7r /2 ~ 8 ~ 1r /2 and r = 1. (b) Write an iterated integral representing the area inside the curve r = 1 and to the right of r = 1/ ( 2 cos 8). Evaluate the integral. (a) Write an iterated integral representing the mass of the disk. (b) Evaluate the integral. 34. Two circular disks, each of radius I, have centers which 16~5 INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES Some double integrals are easier to evaluate in polar, rather than Cartesian, coordinates. Similarly, some triple integrals are easier in non-Cartesian coordinates. Cy!indricaJ Coordinates The cylindrical coordinates of a point (x, y, z) in 3-space are obtained by representing the x andy coordinates in polar coordinates and letting the z-coordinate be the z-coordinate of the Cartesian coordinate system. (See Figure 16.34.) Relation between Cartesian and Cylindrical Coordinates I Each point in 3-space is represented using 0 I s r < oo, 0 S (} S 21r, -oo < z < oo. x = rcos8, y = rsin8, z = z. I I + y2 = r 2 L.~~~-. ~. - . ~--. . . . . . . . . . . ~--~---- . - . . . . . ._._. . . . _. . ~. .-.. . . . . . ._. . . . . . . ~---. --------- As with polar coordinates in the plane, note that x 2 z P = (r, 8, z) ' I I I Z I ,. r8~'~,-,---r----~-,, y I ',, I X (r,8,0) Figure H}-34: Cylindrical coordinates: (r, (}, z) A useful way to visualize cylindrical coordinates is to sketch the surfaces obtained by setting one of the coordinates equal to a constant. See Figures 16.35-16.37