16 MULTIPLE INTEGRALS

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<strong>16</strong> SAMPLE EXAM Problems marked with an asterisk (*) are particularly challenging and should be given careful consideration. 1. Consider the function f (x, y) = x y on the rectangle [1, 2] × [1, 2]. (a) Approximate the value of the integral ∫ 2∫ 2 1 1 f (x, y) dydx by dividing the region into four squares and using the function value at the lower left-hand corner of each square as an approximation for the function value over that square. (b) Does the approximation give an overestimate or an underestimate of the value of the integral? How do you know? 2. Given that ∫ π/2 0 dx 1 + sin 2 x = π 2 √ 2 , (a) evaluate the double integral (b) evaluate the triple integral 3. Consider the rugged region below: ∫ π/2 ∫ π/2 0 0 1 ( 1 + sin 2 x )( 1 + sin 2 ) dx dy y ∫ π/2 ∫ π/2 ∫ ( 1/ 1+sin 2 y ) 0 0 1/ ( 1+sin 2 x ) dzdx dy (a) Divide the region into smaller regions, all of which are Type I. (b) Divide the region into smaller regions, all of which are Type II. 4. Rewrite the integral in rectangular coordinates. ∫ 2π ∫ 1 0 0 932 r 2 dr dθ
Compute ∫∫∫ R 10 dV and ∫∫∫ R xdV. 933 CHAPTER <strong>16</strong> SAMPLE EXAM 5. Evaluate ∫∫ D cos ( x 2 + y 2) dA,whereD = { (x, y) | 1 ≤ x 2 + y 2 ≤ 4 } is a washer with inner radius 1 and outer radius 2. y 2 1 x 6. Consider the ellipse x 2 + 2y 2 = 1. (a) Rewrite this equation in polar coordinates. (b) Write an integral in polar coordinates that gives the area of this ellipse. Note: Your answer will not look simple. 7. Consider the rectangular prism R pictured below:
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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 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: GROUP WORK 2, SECTION 16.9 Transfor
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
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16 MULTIPLE INTEGRALS

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