16 MULTIPLE INTEGRALS

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GROUP WORK 4, SECTION 16.1 Several Ways to Compute Double Integrals Consider the double integral ∫∫ (2 − x − y) dA, where R = [0, 1] × [0, 1]. 1. Estimate the value of the double integral using two equal subdivisions in each direction. 2. Fix x such that 0 ≤ x ≤ 1. What is the area A (x) of the slice shown below? z 2 y=0, z=2-x 1 x 2 A(x) y=1, z=1-x 1 y 3. Find the exact volume of the solid with cross-sectional area A (x) using single variable calculus. 880
16.2 ITERATED INTEGRALS SUGGESTED TIME AND EMPHASIS 1 2 –1 class Essential material POINTS TO STRESS 1. The meaning of ∫ b a ∫ d c f (x, y) dydx for a positive function f (x, y) over a rectangle [a, b] × [c, d]. 2. The statement of Fubini’s Theorem and how it makes computations easier. 3. The geometric meaning of Fubini’s Theorem: slicing the area in two different ways. QUIZ QUESTIONS • Text Question: Consider Figures 1 and 2 in the text. Why is ∫ b a A (x) dx = ∫ d c A (y) dy? z z C b x a x 0 A(x) y x 0 c y d y Answer: The two integrals express the volume of the same solid. ∫ 4 3 x2 ydydx. • Drill Question: Compute ∫ 3 0 Answer: 63 2 MATERIALS FOR LECTURE • Revisit the example ∫∫ ( [−2,2] × [−2,2] x 3 + y 5) dAusing integrated integrals, to illustrate the power of the technique of iteration. • Use an alternate approach to give an intuitive idea of why Fubini’s Theorem is true. Using equal intervals of length x and y in each direction and choosing the lower left corner in each rectangle, we can write the double sum in Definition 16.1.5 as the iterated sum ( ) m∑ n∑ ( ) f xij ∗ , ∑ y∗ ij x y = m n∑ ( ) f xij ∗ , y∗ ij x y i=1 j=1 i=1 j=1 which in the limit gives an iterated integral. Go through some examples such as ∫∫ [−1,3] × [−1,3] xydA and ∫∫ [0,1] × [0,1] (2 − x − y) dAto demonstrate this approach. • Remind the students how, in single-variable calculus, volumes were found by adding up cross-sectional areas. Take the half-cylinder f (x, y) = √ 1 − y 2 ,0≤ x ≤ 2, and find its volume, first by the “old” method, then by expressing it as a double integral. Show how the two techniques are, in essence, the same. 881
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: GROUP WORK 2, SECTION 16.1 An Odd F
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 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 ∫
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(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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