9.3 Theorems of Pappus and Guldinus

Transcription

1 9.3 Theorems of Pappus and Guldinus

2 9.3 Theorems of Pappus and Guldinus Procedures and Strategies, page 1 of 2 Procedures and Strategies for Solving Problems Involving a the Theorems of Pappus and Guldinus To calculate the area of a solid of revolution, 1. identif the ais of revolution, 2. sketch the generating curve, 3. find the distance r C from the ais of revolution to the centroid of the curve, and 4. Appl the formula for the area: a a Curve of length L A = 2 r C L a r C Centroid of curve where L is the length of the curve. b To calculate the volume of a solid of revolution, 1. identif the ais of revolution, 2. sketch the generating area, 3. find the distance r C from the ais of revolution to the centroid of the area, and 4. appl the formula for the volume: V = 2 r C A b b r C Planar region of area A where A is the generating area. Centroid of region b

3 9.3 Theorems of Pappus and Guldinus Procedures and Strategies, page 2 of 2 Notes: 1) If the curve or area is rotated an amount less than a complete revolution, then the factor of 2 in the equations for the volume and area must be replaced b (epressed in radians). 2) If the centroidal distance cannot be found from a table, then ou will have to calculate the distance b evaluating an integral. But ou can save work b noting that ou do not have to calculate r C and L independentl ou need calculate onl the product r C L and that can be found b noting that implies C = el dl dl L C L = el dl So ou onl have to evaluate one integral, namel el dl

4 9.3 Theorems of Pappus and Guldinus Problem Statement for Eample 1 1. Determine the amount of paint required to paint the inside and outside surfaces of the cone, if one gallon of paint covers 300 ft 2. 3 ft 10 ft

5 9.3 Theorems of Pappus and Guldinus Problem Statement for Eample 2 2. Determine the volume of the cone. 3 ft 10 ft

6 9.3 Theorems of Pappus and Guldinus Problem Statement for Eample 3 3. Determine the area of the half-torus (half of a doughnut). 1 m C 4 m z

7 9.3 Theorems of Pappus and Guldinus Problem Statement for Eample 4 4. Determine the volume of the half-torus (half of a doughnut). 1 m 4 m C z

8 9.3 Theorems of Pappus and Guldinus Problem Statement for Eample 5 5. Determine the area of the frustum of the cone. 3 m 4 m O 2 m

9 9.3 Theorems of Pappus and Guldinus Problem Statement for Eample 6 6. Determine the volume of the frustum of the cone. 3 m 4 m O 2 m

10 9.3 Theorems of Pappus and Guldinus Problem Statement for Eample 7 7. Determine the centroidal coordinate rc of a semicircular arc of radius R, given that the area of a sphere of radius R is known to be 4 R 2. r C R C

11 9.3 Theorems of Pappus and Guldinus Problem Statement for Eample 8 8. Determine the centroidal coordinate rc of a semicircular area of radius R, given that the volume of a sphere is known to be (4/3) R 3. Radius = R r C C

12 9.3 Theorems of Pappus and Guldinus Problem Statement for Eample 9 9. A concrete dam is to be constructed in the shape shown. Determine the volume of concrete that would be required. 3.5 m 20 m 40 3 m 1 m 2 m

13 9.3 Theorems of Pappus and Guldinus Problem Statement for Eample The concrete steps shown are in the shape of a quarter circle. Determine the amount of paint required to paint the steps, if one liter of paint covers 1.5 m mm 190 mm 260 mm 190 mm z

14 9.3 Theorems of Pappus and Guldinus Problem Statement for Eample The concrete steps shown are in the shape of a quarter circle. Determine the total number of cubic meters of concrete required to construct the steps. 260 mm 190 mm 260 mm 190 mm z

15 9.3 Theorems of Pappus and Guldinus Problem Statement for Eample Determine the mass of the steel V-belt pulle shown. The densit of the steel is 7840 kg/m mm 12 mm 10 mm 12 mm 4 mm 4 mm 100 mm 70 mm 20 mm 15 mm Front view Side view

16 9.3 Theorems of Pappus and Guldinus Problem Statement for Eample Determine the area of the surface of revolution generated b rotating the curve = z 4, 0 z 1 m, about the z ais. 1 m z

17 9.3 Theorems of Pappus and Guldinus Problem Statement for Eample Determine the volume of the solid of revolution generated b rotating the curve = z 4, 0 z 1 m, about the z ais. 1 m z

18 9.3 Theorems of Pappus and Guldinus Problem Statement for Eample A pharmaceutical compan plans to put a coating 0.01 mm thick on the outside of the pill shown. Determine the amount of coating material required. Radius = 20 mm 7 mm 1.5 mm

19 9.3 Theorems of Pappus and Guldinus Problem Statement for Eample Determine the volume of the funnel. 10 mm 60 mm 5 mm 70 mm 5 mm

20 9.3 Theorems of Pappus and Guldinus Problem Statement for Eample A satellite dish is shaped in the form of a paraboloid of revolution to take advantage of the geometrical fact that all signals traveling parallel to the ais of the paraboloid are reflected through the focus. Determine the amount, in m 2, of reflecting material required to cover the inside surface of the dish. 0.3 m Signals parallel to ais of dish 0.3 m 0.2 m

21 9.3 Theorems of Pappus and Guldinus Problem Statement for Eample Determine the amount of coffee that the coffee mug holds when full to the brim. The radius of the rounded corners and the rim is 15 mm. 80 mm 90 mm

22 9.3 Theorems of Pappus and Guldinus Problem Statement for Eample Determine the capacit of the small bottle of lotion if the bottle is filled half wa up the neck. 5 mm 17.5 mm 15 mm Radius = 20 mm

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