6161103 9.4 theorems of pappus and guldinus

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6161103 9.4 theorems of pappus and guldinus

  1. 1. 9.4 Theorems of Pappus and Guldinus A surface area of revolution is generated by revolving a plane curve about a non-intersecting fixed axis in the plane of the curve A volume of revolution is generated by revolving a plane area bout a nonintersecting fixed axis in the plane of areaExample Line AB is rotated about fixed axis, it generates the surface area of a cone (less area of base)
  2. 2. 9.4 Theorems of Pappus and GuldinusExample Triangular area ABC rotated about the axis would generate the volume of the cone The theorems of Pappus and Guldinus are used to find the surfaces area and volume of any object of revolution provided the generating curves and areas do not cross the axis they are rotated
  3. 3. 9.4 Theorems of Pappus and GuldinusSurface Area Area of a surface of revolution = product of length of the curve and distance traveled by the centroid in generating the surface area A =θ r L
  4. 4. 9.4 Theorems of Pappus and GuldinusVolume Volume of a body of revolution = product of generating area and distance traveled by the centroid in generating the volume V =θ r A
  5. 5. 9.4 Theorems of Pappus and GuldinusComposite Shapes The above two mentioned theorems can be applied to lines or areas that may be composed of a series of composite parts Total surface area or volume generated is the addition of the surface areas or volumes generated by each of the composite parts A = θ ∑(~ L ) r V = θ ∑(~ A) r
  6. 6. 9.4 Theorems of Pappus and GuldinusExample 9.12Show that the surface area of a sphere is A = 4πR2and its volume V = 4/3 πR3
  7. 7. 9.4 Theorems of Pappus and GuldinusSolutionSurface Area Generated by rotating semi-arc about the x axis For centroid, r = 2R / π For surface area, A = θ ~ L; r  2 R  πR = 4πR 2 A = 2π   π 
  8. 8. 9.4 Theorems of Pappus and GuldinusSolutionVolume Generated by rotating semicircular area about the x axis For centroid, r = 4 R / 3π For volume, V = θ ~ A; r  4 R   1 πR 2  = 4 πR3 V = 2π      3π   2  3

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