16 MULTIPLE INTEGRALS

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CHAPTER 16 MULTIPLE INTEGRALS WORKSHOP/DISCUSSION • Have the students work several examples, such as ∫∫ [0,1] × [0,1] y√ 1 − x 2 sin ( 2πy 2) dA, which can be computed as ∫ 1 0 f (x) dx ∫ 1 0 g (y) dy. • Find the volume of the solids described by ∫∫ ( [−1,1] × [−1,1] 1 + x 2 + y 2) dA. ∫∫ [ ( − √ 2, √ ] 2 − x 2 ) dA and 2 × [−2,3] GROUP WORK 1: Regional Differences If the students get stuck on this one, give them the hint that Problem 1(b) can be done by finding the double integral over the square, and then using the symmetry of the function to compute the area over R. The regions in the remaining problems can be broken into rectangles. Answers: 1.(a) 3 4 (b) 1 2 2.(a) −8 (b)−5 (c)− 9 2 GROUP WORK 2: Practice with Double Integrals It is a good idea to give the students some practice with straightforward computations of the type included in Problem 1. It is advised to put the students in pairs or have them work individually, as opposed to putting them in larger groups. The students are not to actually compute the integral in Problem 2; rather, they should recognize that each slice integrates to zero. Think about what happens when integrating with respect to x first. Answers: 1. (a) 24 √ √ √ √ 5 6 − 10 3 5 − 6 5 3 + 8 15 2 (b) − 3 2 ln 3 + 1 2 + 2 ln 2 (c) e2 − 1 − e 2. True GROUP WORK 3: The Shape of the Solid In Problem 3, the students could first change the order of integration to more easily recognize the “pup tent” shape of the resulting solid. Answers: 1. Irregular tetrahedron, volume 3 2 2. Half a cylinder, volume 12π 3. “Pup tent”, volume 5 HOMEWORK PROBLEMS Core Exercises: 1, 3, 6, 17, 23, 26, 32, 35 Sample Assignment: 1, 2, 3, 6, 9, 13, 15, 17, 20, 23, 26, 28, 32, 35 Exercise D A N G 1 × 2 × 3 × 6 × 9 × 13 × 15 × Exercise D A N G 20 × 23 × 26 × 28 × 32 × × 35 × 882
GROUP WORK 1, SECTION 16.2 Regional Differences 1. Calculate the double integral ∫∫ R (x + y) dAfor the following regions R: (a) y (b) y 1 1 _ 2 R 1 R 0 1 _ 2 1 x 0 1 x 2. Calculate the double integral ∫∫ ( R xy − y 3 ) dAfor the following regions R. (a) y (b) y (c) y 2 2 2 R 1 R 1 R 1 R _1 0 1 x _1 0 1 x _1 0 1 x 883
Page 1 and 2: 16 MULTIPLE INTEGRALS 16.1 DOUBLE I
Page 3 and 4: SECTION 16.1 DOUBLE INTEGRALS OVER
Page 5 and 6: GROUP WORK 2, SECTION 16.1 An Odd F
Page 7 and 8: GROUP WORK 2, SECTION 16.1 An Odd F
Page 9: 16.2 ITERATED INTEGRALS SUGGESTED T
Page 13 and 14: GROUP WORK 3, SECTION 16.2 The Shap
Page 15 and 16: SECTION 16.3 DOUBLE INTEGRALS OVER
Page 17 and 18: GROUP WORK 1, SECTION 16.3 TypeIorT
Page 19 and 20: GROUP WORK 3, SECTION 16.3 Bounding
Page 21 and 22: SECTION 16.4 DOUBLE INTEGRALS IN PO
Page 23 and 24: SECTION 16.4 DOUBLE INTEGRALS IN PO
Page 25 and 26: GROUP WORK 2, SECTION 16.4 Fun with
Page 27 and 28: 16.5 APPLICATIONS OF DOUBLE INTEGRA
Page 29 and 30: SECTION 16.5 APPLICATIONS OF DOUBLE
Page 31 and 32: GROUP WORK 3, SECTION 16.5 A Slick
Page 33 and 34: SECTION 16.6 TRIPLE INTEGRALS • S
Page 35 and 36: ( ∫ 1 ∫ √ 4−x 2 0 1 SECTION
Page 37 and 38: GROUP WORK 2, SECTION 16.6 Setting
Page 39 and 40: DISCOVERY PROJECT Volumes of Hypers
Page 41 and 42: SECTION 16.7 TRIPLE INTEGRALS IN CY
Page 43 and 44: GROUP WORK, SECTION 16.7 A Partiall
Page 45 and 46: 16.8 TRIPLE INTEGRALS IN SPHERICAL
Page 47 and 48: SECTION 16.8 TRIPLE INTEGRALS IN SP
Page 49 and 50: GROUP WORK 1, SECTION 16.8 Describe
Page 51 and 52: GROUP WORK 2, SECTION 16.8 Setting
Page 53 and 54: APPLIED PROJECT Roller Derby This p
Page 55 and 56: SECTION 16.9 CHANGE OF VARIABLES IN
Page 57 and 58: SECTION 16.9 CHANGE OF VARIABLES IN
Page 59 and 60: GROUP WORK 2, SECTION 16.9 Transfor
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Compute ∫∫∫ R 10 dV and ∫
Page 63 and 64:
(b) Evaluate ∫∫ Pac-Man xdAand
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(b) 3. (a) CHAPTER 16 SAMPLE EXAM S
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CHAPTER 16 SAMPLE EXAM SOLUTIONS (a
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16 MULTIPLE INTEGRALS

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