16 MULTIPLE INTEGRALS

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CHAPTER 16 MULTIPLE INTEGRALS • Present some different uses of polar integrals: 1. Compute ∫∫ ( R x 2 + y 2) 2 dA,whereR is the region enclosed by x 2 + y 2 = 4 between the lines y = x and y =−x. y 2 _2 _1 0 1 2 x _2 2. Compute the volume between the cone z 2 = x 2 + y 2 and the paraboloid z = 4 − x 2 − y 2 . GROUPWORK1:ThePolarAreaFormula Perhaps start by having students guess the final answer to Problem 1. Notice that the students will have to be able to antidifferentiate cos 2 θ and sin 2 θ to complete this activity. Answers: 1. 2 ∫ π 0 1 2 (1 + cos θ)2 dθ = 3π 2 2. ∫ π 0 [ 12 (1 + sin θ) 2 − 1 2 (1)2] dθ = π 4 + 2 GROUP WORK 2: Fun with Polar Area Many students will write ∫ 3π ∫ θ 0 0 rdrdθ. Try to get them to see for themselves that they must now subtract the area of the region that is counted twice in this expression. Answers: 1. 9 2 π3 − 19 6 π3 = 4 3 π3 2. ∫ π 0 ∫ 2 0 e−r 2 rdrdθ = π ( 2 1 − e −4 ) GROUP WORK 3: Fun with Polar Volume If the students get stuck on Problem 2, point out that the intersection can be shown to be x 2 + y 2 = 3 4 , z = 1 2 , and hence the integral is over the region x 2 + y 2 ≤ 3 4 . If time is an issue, the students can be instructed to set up the integrals without solving them. Answers: 1. ∫ 2π 0 2. ∫ 2π 0 ∫ 1 (√ 0 1 − r 2 + 1) rdrdθ = 5π 3 ∫ 3/4 [√ 0 1 − r 2 − (1 − √ )] ( 1 − r 2 rdrdθ = 48 π 37 − 7 √ ) 7 894
SECTION 16.4 DOUBLE INTEGRALS IN POLAR COORDINATES HOMEWORK PROBLEMS Core Exercises: 2, 5, 11, 15, 22, 30, 33, 35 Sample Assignment: 2, 5, 8, 11, 15, 17, 20, 22, 25, 30, 31, 33, 35 Exercise D A N G 2 × × 5 × × 8 × 11 × 15 × 17 × 20 × 22 × 25 × 30 × 31 × 33 × × 35 × 895
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 and 10: 16.2 ITERATED INTEGRALS SUGGESTED T
Page 11 and 12: GROUP WORK 1, SECTION 16.2 Regional
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: 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
Page 61 and 62: Compute ∫∫∫ R 10 dV and ∫
Page 63 and 64: (b) Evaluate ∫∫ Pac-Man xdAand
Page 65 and 66: (b) 3. (a) CHAPTER 16 SAMPLE EXAM S
Page 67: CHAPTER 16 SAMPLE EXAM SOLUTIONS (a

16 MULTIPLE INTEGRALS

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