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- 1. Cylindrical and Spherical Coordinates
- 2. The cylindrical coordinate system is the combinationof using polar coordinates for points in the xy–planewith z as the 3rdcoordinate.Cylindrical and Spherical Coordinates
- 3. Example A. a. Plot the point (3,120o, 4)in cylindrical coordinate. Convert it torectangular coordinate.The cylindrical coordinate system is the combinationof using polar coordinates for points in the xy–planewith z as the 3rdcoordinate.Cylindrical and Spherical Coordinates
- 4. Example A. a. Plot the point (3,120o, 4)in cylindrical coordinate. Convert it torectangular coordinate.3120oxyzThe cylindrical coordinate system is the combinationof using polar coordinates for points in the xy–planewith z as the 3rdcoordinate.Cylindrical and Spherical Coordinates
- 5. Example A. a. Plot the point (3,120o, 4)in cylindrical coordinate. Convert it torectangular coordinate.3120o4xyz(3, 120o, 4)The cylindrical coordinate system is the combinationof using polar coordinates for points in the xy–planewith z as the 3rdcoordinate.Cylindrical and Spherical Coordinates
- 6. Example A. a. Plot the point (3,120o, 4)in cylindrical coordinate. Convert it torectangular coordinate.3120o4xyz(3, 120o, 4)x = 3cos(120o) = –3/2y = 3sin(120o) = √3Hence the point is (–3/2, √3, 4)The cylindrical coordinate system is the combinationof using polar coordinates for points in the xy–planewith z as the 3rdcoordinate.Cylindrical and Spherical Coordinates
- 7. Example A. a. Plot the point (3,120o, 4)in cylindrical coordinate. Convert it torectangular coordinate.3120o4xyz(3, 120o, 4)x = 3cos(120o) = –3/2y = 3sin(120o) = √3Hence the point is (–3/2, √3, 4)b. Convert (3, –3, 1) into tocylindrical coordinate.The cylindrical coordinate system is the combinationof using polar coordinates for points in the xy–planewith z as the 3rdcoordinate.Cylindrical and Spherical Coordinates
- 8. Example A. a. Plot the point (3,120o, 4)in cylindrical coordinate. Convert it torectangular coordinate.3120o4xyz(3, 120o, 4)x = 3cos(120o) = –3/2y = 3sin(120o) = √3Hence the point is (–3/2, √3, 4)b. Convert (3, –3, 1) into tocylindrical coordinate.The cylindrical coordinate system is the combinationof using polar coordinates for points in the xy–planewith z as the 3rdcoordinate.Cylindrical and Spherical Coordinates
- 9. Example A. a. Plot the point (3,120o, 4)in cylindrical coordinate. Convert it torectangular coordinate.3120o4xyz(3, 120o, 4)x = 3cos(120o) = –3/2y = 3sin(120o) = √3Hence the point is (–3/2, √3, 4)b. Convert (3, –3, 1) into tocylindrical coordinate.θ = 315o, r = √9 + 9 = √18Hence the point is (√18, 315o,1) the cylindrical coordinate.The cylindrical coordinate system is the combinationof using polar coordinates for points in the xy–planewith z as the 3rdcoordinate.Cylindrical and Spherical Coordinates
- 10. Example A. a. Plot the point (3,120o, 4)in cylindrical coordinate. Convert it torectangular coordinate.3120o4xyz(3, 120o, 4)x = 3cos(120o) = –3/2y = 3sin(120o) = √3Hence the point is (–3/2, √3, 4)b. Convert (3, –3, 1) into tocylindrical coordinate.(√18, 315o, 0)θ = 315o, r = √9 + 9 = √18Hence the point is (√18, 315o,1) the cylindrical coordinate. xyzThe cylindrical coordinate system is the combinationof using polar coordinates for points in the xy–planewith z as the 3rdcoordinate.Cylindrical and Spherical Coordinates
- 11. Example A. a. Plot the point (3,120o, 4)in cylindrical coordinate. Convert it torectangular coordinate.3120o4xyz(3, 120o, 4)x = 3cos(120o) = –3/2y = 3sin(120o) = √3Hence the point is (–3/2, √3, 4)b. Convert (3, –3, 1) into tocylindrical coordinate.(√18, 315o, 0)θ = 315o, r = √9 + 9 = √18Hence the point is (√18, 315o,1) the cylindrical coordinate. xyz(√18, 315o, 1) = (3, –3, 1)1The cylindrical coordinate system is the combinationof using polar coordinates for points in the xy–planewith z as the 3rdcoordinate.Cylindrical and Spherical Coordinates
- 12. Cylindrical and Spherical CoordinatesThe constant equationsr = k describes thecylinder of radius k, thusthe name "cylindricalcoordinate".Example: Sketch r = 2
- 13. Cylindrical and Spherical CoordinatesThe constant equationsr = k describes thecylinder of radius k, thusthe name "cylindricalcoordinate".Example: Sketch r = 22
- 14. Cylindrical and Spherical CoordinatesThe constant equationsr = k describes thecylinder of radius k, thusthe name "cylindricalcoordinate".Example: Sketch r = 22The constant equationsθ = k describes thevertical half plane throughthe origin, at the angle kwith x-axis. (r > 0)
- 15. Cylindrical and Spherical CoordinatesThe constant equationsr = k describes thecylinder of radius k, thusthe name "cylindricalcoordinate".Example: Sketch r = 22The constant equationsθ = k describes thevertical half plane throughthe origin, at the angle kwith x-axis. (r > 0)Example: Sketch θ =3π/4
- 16. Cylindrical and Spherical CoordinatesThe constant equationsr = k describes thecylinder of radius k, thusthe name "cylindricalcoordinate".XYExample: Sketch r = 22The constant equationsθ = k describes thevertical half plane throughthe origin, at the angle kwith x-axis. (r > 0)Example: Sketch θ =3π/43π/4Y
- 17. Cylindrical and Spherical CoordinatesIn spherical coordinate, a point in space isrepresented by (ρ, θ, φ) where ρ gives the distancefrom the origin to the point with θ and φ as shown inthe figure.θXYZφ ρ
- 18. Cylindrical and Spherical CoordinatesIn spherical coordinate, a point in space isrepresented by (ρ, θ, φ) where ρ gives the distancefrom the origin to the point with θ and φ as shown inthe figure. We restrict ρ > 0, 2π > θ > 0, andπ > φ > 0.θXYZφ ρ
- 19. Cylindrical and Spherical CoordinatesIn spherical coordinate, a point in space isrepresented by (ρ, θ, φ) where ρ gives the distancefrom the origin to the point with θ and φ as shown inthe figure. We restrict ρ > 0, 2π > θ > 0, andπ > φ > 0.θXYZφ ρr = ρsin(φ)
- 20. Cylindrical and Spherical CoordinatesIn spherical coordinate, a point in space isrepresented by (ρ, θ, φ) where ρ gives the distancefrom the origin to the point with θ and φ as shown inthe figure. We restrict ρ > 0, 2π > θ > 0, andπ > φ > 0.θXYZφ ρr = ρsin(φ)z = ρcos(φ)
- 21. Cylindrical and Spherical CoordinatesIn spherical coordinate, a point in space isrepresented by (ρ, θ, φ) where ρ gives the distancefrom the origin to the point with θ and φ as shown inthe figure. We restrict ρ > 0, 2π > θ > 0, andπ > φ > 0.θXYZThe main conversion rule thatconnects all these system is thatr = ρsin(φ). φ ρr = ρsin(φ)z = ρcos(φ)
- 22. Cylindrical and Spherical CoordinatesIn spherical coordinate, a point in space isrepresented by (ρ, θ, φ) where ρ gives the distancefrom the origin to the point with θ and φ as shown inthe figure. We restrict ρ > 0, 2π > θ > 0, andπ > φ > 0.θXYZThe main conversion rule thatconnects all these system is thatr = ρsin(φ).Hence x = ρsin(φ)cos(θ),φ ρr = ρsin(φ)z = ρcos(φ)
- 23. Cylindrical and Spherical CoordinatesIn spherical coordinate, a point in space isrepresented by (ρ, θ, φ) where ρ gives the distancefrom the origin to the point with θ and φ as shown inthe figure. We restrict ρ > 0, 2π > θ > 0, andπ > φ > 0.θXYZThe main conversion rule thatconnects all these system is thatr = ρsin(φ).Hence x = ρsin(φ)cos(θ),y = ρsin(φ)sin(θ),φ ρr = ρsin(φ)z = ρcos(φ)
- 24. Cylindrical and Spherical CoordinatesIn spherical coordinate, a point in space isrepresented by (ρ, θ, φ) where ρ gives the distancefrom the origin to the point with θ and φ as shown inthe figure. We restrict ρ > 0, 2π > θ > 0, andπ > φ > 0.θXYZThe main conversion rule thatconnects all these system is thatr = ρsin(φ).Hence x = ρsin(φ)cos(θ),y = ρsin(φ)sin(θ),z = ρcos(φ)φ ρr = ρsin(φ)z = ρcos(φ)
- 25. Cylindrical and Spherical CoordinatesIn spherical coordinate, a point in space isrepresented by (ρ, θ, φ) where ρ gives the distancefrom the origin to the point with θ and φ as shown inthe figure. We restrict ρ > 0, 2π > θ > 0, andπ > φ > 0.θXYZThe main conversion rule thatconnects all these system is thatr = ρsin(φ).Hence x = ρsin(φ)cos(θ),y = ρsin(φ)sin(θ),z = ρcos(φ)ρ = √x2+ y2+ z2, tan(φ) = r/zφ ρr = ρsin(φ)z = ρcos(φ)
- 26. Cylindrical and Spherical CoordinatesExample: Plot the point (3, 2π/3, π/4) in cylindricalcoordinate. Convert it to the other systems.
- 27. Cylindrical and Spherical CoordinatesExample: Plot the point (3, 2π/3, π/4) in cylindricalcoordinate. Convert it to the other systems.2π/33XYZπ/4
- 28. Cylindrical and Spherical CoordinatesExample: Plot the point (3, 2π/3, π/4) in cylindricalcoordinate. Convert it to the other systems.r = 3sin(π/4)2π/33XYZπ/4
- 29. Cylindrical and Spherical CoordinatesExample: Plot the point (3, 2π/3, π/4) in cylindricalcoordinate. Convert it to the other systems.r = 3sin(π/4) = (3√2)/22π/33XYZπ/4
- 30. Cylindrical and Spherical CoordinatesExample: Plot the point (3, 2π/3, π/4) in cylindricalcoordinate. Convert it to the other systems.r = 3sin(π/4) = (3√2)/2x = rcos(θ)2π/33XYZπ/4
- 31. Cylindrical and Spherical CoordinatesExample: Plot the point (3, 2π/3, π/4) in cylindricalcoordinate. Convert it to the other systems.r = 3sin(π/4) = (3√2)/2x = rcos(θ) = (3√2)/2 * cos(2π/3)2π/33XYZπ/4
- 32. Cylindrical and Spherical CoordinatesExample: Plot the point (3, 2π/3, π/4) in cylindricalcoordinate. Convert it to the other systems.r = 3sin(π/4) = (3√2)/2x = rcos(θ) = (3√2)/2 * cos(2π/3)= (-3√2)/42π/33XYZπ/4
- 33. Cylindrical and Spherical CoordinatesExample: Plot the point (3, 2π/3, π/4) in cylindricalcoordinate. Convert it to the other systems.r = 3sin(π/4) = (3√2)/2x = rcos(θ) = (3√2)/2 * cos(2π/3)= (-3√2)/4y = rsin(θ) = 3√2/2 * sin(2π/3)2π/33XYZπ/4
- 34. Cylindrical and Spherical CoordinatesExample: Plot the point (3, 2π/3, π/4) in cylindricalcoordinate. Convert it to the other systems.r = 3sin(π/4) = (3√2)/2x = rcos(θ) = (3√2)/2 * cos(2π/3)= (-3√2)/4y = rsin(θ) = 3√2/2 * sin(2π/3)= (3√6)/42π/33XYZπ/4
- 35. Cylindrical and Spherical CoordinatesExample: Plot the point (3, 2π/3, π/4) in cylindricalcoordinate. Convert it to the other systems.r = 3sin(π/4) = (3√2)/2x = rcos(θ) = (3√2)/2 * cos(2π/3)= (-3√2)/4y = rsin(θ) = 3√2/2 * sin(2π/3)= (3√6)/4z = 3cos(π/4) = (3√3)/2 2π/33XYZπ/4
- 36. Cylindrical and Spherical CoordinatesExample: Plot the point (3, 2π/3, π/4) in cylindricalcoordinate. Convert it to the other systems.r = 3sin(π/4) = (3√2)/2x = rcos(θ) = (3√2)/2 * cos(2π/3)= (-3√2)/4y = rsin(θ) = 3√2/2 * sin(2π/3)= (3√6)/4z = 3cos(π/4) = (3√3)/2Hence the point is:((-3√2)/4, (3√6)/4, (3√3)/2)in the rectangular coordinate;2π/33XYZπ/4
- 37. Cylindrical and Spherical CoordinatesExample: Plot the point (3, 2π/3, π/4) in cylindricalcoordinate. Convert it to the other systems.r = 3sin(π/4) = (3√2)/2x = rcos(θ) = (3√2)/2 * cos(2π/3)= (-3√2)/4y = rsin(θ) = 3√2/2 * sin(2π/3)= (3√6)/4z = 3cos(π/4) = (3√3)/2Hence the point is:((-3√2)/4, (3√6)/4, (3√3)/2)in the rectangular coordinate;((3√2)/2, 2π/3, (3√3)/2) inthe cylindrical coordinate.2π/33XYZπ/4
- 38. Cylindrical and Spherical CoordinatesThe constant equationsρ = k describe the sphereof radius k, thus the name"spherical coordinate".Example: Sketch ρ = 2
- 39. Cylindrical and Spherical CoordinatesThe constant equationsρ = k describe the sphereof radius k, thus the name"spherical coordinate".Example: Sketch ρ = 2 Example: Sketch φ = π/2φ = π/2The constant equationsφ = k describe the cone ofof angle φ with the z axis.
- 40. Cylindrical and Spherical CoordinatesExample: Convert the equation z = √k2– x2– y2intothe polar equation and the spherical equation.
- 41. Cylindrical and Spherical CoordinatesExample: Convert the equation z = √k2– x2– y2intothe polar equation and the spherical equation.k
- 42. Cylindrical and Spherical CoordinatesExample: Convert the equation z = √k2– x2– y2intothe polar equation and the spherical equation.Since r = x2+ y2,k
- 43. Cylindrical and Spherical CoordinatesExample: Convert the equation z = √k2– x2– y2intothe polar equation and the spherical equation.Since r = x2+ y2, the polarequation is z = √k2– r2k
- 44. Cylindrical and Spherical CoordinatesExample: Convert the equation z = √k2– x2– y2intothe polar equation and the spherical equation.Since r = x2+ y2, the polarequation is z = √k2– r2For spherical equation, we getwith 0 < φ < π/2ρ = kk
- 45. Cylindrical and Spherical CoordinatesExample: Convert the equation z = x2+ y2into thepolar equation and the spherical equation.
- 46. Cylindrical and Spherical CoordinatesExample: Convert the equation z = x2+ y2into thepolar equation and the spherical equation.
- 47. Cylindrical and Spherical CoordinatesExample: Convert the equation z = x2+ y2into thepolar equation and the spherical equation.Since r = √x2+ y2, the polarequation is z = r2
- 48. Cylindrical and Spherical CoordinatesExample: Convert the equation z = x2+ y2into thepolar equation and the spherical equation.Since r = √x2+ y2, the polarequation is z = r2rz = r2
- 49. Cylindrical and Spherical CoordinatesExample: Convert the equation z = x2+ y2into thepolar equation and the spherical equation.Since r = √x2+ y2, the polarequation is z = r2For spherical equation, setz = ρcos(φ)r = ρsin(φ),rz = r2ρφ
- 50. Cylindrical and Spherical CoordinatesExample: Convert the equation z = x2+ y2into thepolar equation and the spherical equation.Since r = √x2+ y2, the polarequation is z = r2For spherical equation, setz = ρcos(φ)r = ρsin(φ),We get= (ρsin(φ))2ρcos(φ)rz = r2ρφ
- 51. Cylindrical and Spherical CoordinatesExample: Convert the equation z = x2+ y2into thepolar equation and the spherical equation.Since r = √x2+ y2, the polarequation is z = r2For spherical equation, setz = ρcos(φ)r = ρsin(φ),We get= (ρsin(φ))2ρcos(φ)= ρ*sin2(φ)cos(φ)or rz = r2ρφ
- 52. Cylindrical and Spherical CoordinatesExample: Convert the equation z = x2+ y2into thepolar equation and the spherical equation.Since r = √x2+ y2, the polarequation is z = r2For spherical equation, setz = ρcos(φ)r = ρsin(φ),We get= (ρsin(φ))2ρcos(φ)= ρ*sin2(φ)cos(φ)or rz = r2ρφ
- 53. Triple Integrals in Cylindrical and Spherical Coordinates
- 54. Triple Integrals in Cylindrical and Spherical CoordinatesA solids V may be represented easier in cylindricalor sphereical format than in rectangularcoordinates.
- 55. Triple Integrals in Cylindrical and Spherical CoordinatesA solids V may be represented easier in cylindricalor sphereical format than in rectangularcoordinates. Thus a triple integral of a function withdomain V might be easier to compute if we utilizecylindrical or spherical coordinates.
- 56. Triple Integrals in Cylindrical and Spherical CoordinatesA solids V may be represented easier in cylindricalor sphereical format than in rectangularcoordinates. Thus a triple integral of a function withdomain V might be easier to compute if we utilizecylindrical or spherical coordinates.Let w = h(r, θ, z ) be a function overV in cylindrical coordinates.Cylindrical Coordinates
- 57. Triple Integrals in Cylindrical and Spherical CoordinatesLet w = h(r, θ, z ) be a function overV in cylindrical coordinates.We partion V by small incrementsof Δr, Δθ, and Δz.Cylindrical CoordinatesA solids V may be represented easier in cylindricalor sphereical format than in rectangularcoordinates. Thus a triple integral of a function withdomain V might be easier to compute if we utilizecylindrical or spherical coordinates.
- 58. Triple Integrals in Cylindrical and Spherical CoordinatesLet w = h(r, θ, z ) be a function overV in cylindrical coordinates.We partion V by small incrementsof Δr, Δθ, and Δz.Cylindrical CoordinatesΔθXYZΔrΔrΔzA solids V may be represented easier in cylindricalor sphereical format than in rectangularcoordinates. Thus a triple integral of a function withdomain V might be easier to compute if we utilizecylindrical or spherical coordinates.
- 59. Triple Integrals in Cylindrical and Spherical CoordinatesLet w = h(r, θ, z ) be a function overV in cylindrical coordinates.We partion V by small incrementsof Δr, Δθ, and Δz. A typicalΔV=ΔrxΔθxΔ is a cylindrical wedgewhose volume depends on r, i.eits distance to the z-axis.Cylindrical CoordinatesΔθXYZΔrΔrΔzA solids V may be represented easier in cylindricalor sphereical format than in rectangularcoordinates. Thus a triple integral of a function withdomain V might be easier to compute if we utilizecylindrical or spherical coordinates.
- 60. Triple Integrals in Cylindrical and Spherical CoordinatesIn particular, let r*be thedistance to the center ofΔrxΔθ,ΔθXYZΔrΔzr*
- 61. Triple Integrals in Cylindrical and Spherical CoordinatesIn particular, let r*be thedistance to the center ofΔrxΔθ, then volume ofΔV = r*ΔrΔθΔz since thebase area is r*ΔrΔθ. ΔθXYZΔrΔzr*
- 62. Triple Integrals in Cylindrical and Spherical CoordinatesIn particular, let r*be thedistance to the center ofΔrxΔθ, then volume ofΔV = r*ΔrΔθΔz since thebase area is r*ΔrΔθ. ΔθXYZΔrΔzr*Let (ri, θi, zi) is the center point of each wedge in the partition,(ri, θi, zi)
- 63. Triple Integrals in Cylindrical and Spherical CoordinatesIn particular, let r*be thedistance to the center ofΔrxΔθ, then volume ofΔV = r*ΔrΔθΔz since thebase area is r*ΔrΔθ. ΔθXYZΔrΔzr*= lim Σh(ri, θi, zi) ri ΔrΔθΔz,Δr,Δθ,Δz0then h(r, θ, z) dV∫ ∫∫vLet (ri, θi, zi) is the center point of each wedge in the partition,(ri, θi, zi)
- 64. Triple Integrals in Cylindrical and Spherical CoordinatesIn particular, let r*be thedistance to the center ofΔrxΔθ, then volume ofΔV = r*ΔrΔθΔz since thebase area is r*ΔrΔθ. ΔθXYZΔrΔzr*= lim Σh(ri, θi, zi) ri ΔrΔθΔz,Δr,Δθ,Δz0then h(r, θ, z) dV∫ ∫∫vLet (ri, θi, zi) is the center point of each wedge in the partition,which may be reformulated as an integral over drdθdzdepending on the description of V in polar coordinates.(ri, θi, zi)
- 65. Triple Integrals in Cylindrical and Spherical CoordinatesIn particular, let r*be thedistance to the center ofΔrxΔθ, then volume ofΔV = r*ΔrΔθΔz since thebase area is r*ΔrΔθ. ΔθXYZΔrΔzr*= lim Σh(ri, θi, zi) ri ΔrΔθΔz,Δr,Δθ,Δz0then h(r, θ, z) dV∫ ∫∫vLet (ri, θi, zi) is the center point of each wedge in the partition,which may be reformulated as an integral over drdθdzdepending on the description of V in polar coordinates.Usually V is given as {D; g(r, θ) < z < f(r, θ)} where D is thepolar representation of the base of V in the xy-plane, withg(r, θ) and f(r, θ) form the floor and ceiling of V.(ri, θi, zi)
- 66. Triple Integrals in Cylindrical and Spherical CoordinatesXYZπ/6π/6357VExample: Write V usingcylindrical coordinates.
- 67. Triple Integrals in Cylindrical and Spherical CoordinatesXYZπ/6π/6357VExample: Write V usingcylindrical coordinates.In cylindrical coordinate, V is{π/6 < θ < π/3;
- 68. Triple Integrals in Cylindrical and Spherical CoordinatesXYZπ/6π/6357VExample: Write V usingcylindrical coordinates.In cylindrical coordinate, V is{π/6 < θ < π/3; 5 < r < 7;
- 69. Triple Integrals in Cylindrical and Spherical CoordinatesXYZπ/6π/6357VExample: Write V usingcylindrical coordinates.In cylindrical coordinate, V is{π/6 < θ < π/3; 5 < r < 7; 0 < z < 3}or in any order of r, θ, and z.
- 70. Triple Integrals in Cylindrical and Spherical CoordinatesXYZπ/6π/6357VExample: Write V usingcylindrical coordinates.In cylindrical coordinate, V is{π/6 < θ < π/3; 5 < r < 7; 0 < z < 3}or in any order of r, θ, and z.Example: Write V usingcylindrical coordinates.z=√25–x2–y233VD15
- 71. Triple Integrals in Cylindrical and Spherical CoordinatesXYZπ/6π/6357VExample: Write V usingcylindrical coordinates.In cylindrical coordinate, V is{π/6 < θ < π/3; 5 < r < 7; 0 < z < 3}or in any order of r, θ, and z.Example: Write V usingcylindrical coordinates.The projection D of V onto thexy-coordinate is{0 < θ < 2π; 0 < r < 3}.z=√25–x2–y233VD15
- 72. Triple Integrals in Cylindrical and Spherical CoordinatesXYZπ/6π/6357VExample: Write V usingcylindrical coordinates.In cylindrical coordinate, V is{π/6 < θ < π/3; 5 < r < 7; 0 < z < 3}or in any order of r, θ, and z.Example: Write V usingcylindrical coordinates.The projection D of V onto thexy-coordinate is{0 < θ < 2π; 0 < r < 3}.z=√25–x2–y233VD15z = √25 – x2– y2is z = √25 – r2
- 73. Triple Integrals in Cylindrical and Spherical CoordinatesXYZπ/6π/6357VExample: Write V usingcylindrical coordinates.In cylindrical coordinate, V is{π/6 < θ < π/3; 5 < r < 7; 0 < z < 3}or in any order of r, θ, and z.Example: Write V usingcylindrical coordinates.The projection D of V onto thexy-coordinate is{0 < θ < 2π; 0 < r < 3}.z=√25–x2–y233VD15z = √25 – x2– y2is z = √25 – r2Hence V is {0 < θ < 2π; 0 < r < 3; 1 < z < √25 – r2}.
- 74. Triple Integrals in Cylindrical and Spherical Coordinates=then h(r, θ, z) dV∫ ∫∫vTheorem: Given a solid V in cylindrical coordinate as{A < θ < B; g(θ) < r < f(θ); G(r, θ) < z < F(r, θ) }h(r, θ, z) r dz dr dθ∫ ∫∫Bθ=A r=g(θ)f(θ)z=G(r,θ)F(r,θ)In paricular if h(r, θ, z) = 1, we obtain the volume of V.
- 75. Triple Integrals in Cylindrical and Spherical Coordinates=then h(r, θ, z) dV∫ ∫∫vTheorem: Given a solid V in cylindrical coordinate as{A < θ < B; g(θ) < r < f(θ); G(r, θ) < z < F(r, θ) }h(r, θ, z) r dz dr dθ∫ ∫∫Bθ=A r=g(θ)f(θ)z=G(r,θ)F(r,θ)In paricular if h(r, θ, z) = 1, we obtain the volume of V.Example: Find the volume of Vusing cylindrical coordinates.z=√25–x2–y233VD15
- 76. Triple Integrals in Cylindrical and Spherical Coordinates=then h(r, θ, z) dV∫ ∫∫vTheorem: Given a solid V in cylindrical coordinate as{A < θ < B; g(θ) < r < f(θ); G(r, θ) < z < F(r, θ) }h(r, θ, z) r dz dr dθ∫ ∫∫Bθ=A r=g(θ)f(θ)z=G(r,θ)F(r,θ)In paricular if h(r, θ, z) = 1, we obtain the volume of V.Example: Find the volume of Vusing cylindrical coordinates.z=√25–x2–y233VD15Since V is{0 < θ < 2π; 0 < r < 3; 1 < z < √25 – r2}.
- 77. Triple Integrals in Cylindrical and Spherical Coordinates=then h(r, θ, z) dV∫ ∫∫vTheorem: Given a solid V in cylindrical coordinate as{A < θ < B; g(θ) < r < f(θ); G(r, θ) < z < F(r, θ) }h(r, θ, z) r dz dr dθ∫ ∫∫Bθ=A r=g(θ)f(θ)z=G(r,θ)F(r,θ)In paricular if h(r, θ, z) = 1, we obtain the volume of V.Example: Find the volume of Vusing cylindrical coordinates.z=√25–x2–y233VD15Since V is{0 < θ < 2π; 0 < r < 3; 1 < z < √25 – r2}.Therefore its volume is1* r dz dr dθ∫ ∫∫2πθ=0 r=03z=1√25 – r2
- 78. Triple Integrals in Cylindrical and Spherical Coordinates1* r dz dr dθ∫ ∫∫2πθ=0 r=03z=1√25 – r2
- 79. Triple Integrals in Cylindrical and Spherical Coordinates1* r dz dr dθ∫ ∫∫2πθ=0 r=03z=1√25 – r2= rz | dr dθ∫ ∫2πθ=0 r=03z=1√25 – r2
- 80. Triple Integrals in Cylindrical and Spherical Coordinates1* r dz dr dθ∫ ∫∫2πθ=0 r=03z=1√25 – r2= rz | dr dθ∫ ∫2πθ=0 r=03z=1√25 – r2= r(25 – r2)1/2– r dr dθ∫ ∫2πθ=0 r=03
- 81. Triple Integrals in Cylindrical and Spherical Coordinates1* r dz dr dθ∫ ∫∫2πθ=0 r=03z=1√25 – r2= rz | dr dθ∫ ∫2πθ=0 r=03z=1√25 – r2= r(25 – r2)1/2– r dr dθ∫ ∫2πθ=0 r=03= – (25 – r2)3/2– ½ r2| dθ∫2πθ=0 r=0331
- 82. Triple Integrals in Cylindrical and Spherical Coordinates1* r dz dr dθ∫ ∫∫2πθ=0 r=03z=1√25 – r2= rz | dr dθ∫ ∫2πθ=0 r=03z=1√25 – r2= r(25 – r2)1/2– r dr dθ∫ ∫2πθ=0 r=03= – (25 – r2)3/2– ½ r2| dθ∫2πθ=0 r=0331= dθ∫2πθ=0695=395π
- 83. Triple Integrals in Cylindrical and Spherical CoordinatesLet w = h(ρ, θ, φ ) be a functionover V in sphereical coordinates.Spherical Coordinates
- 84. Triple Integrals in Cylindrical and Spherical CoordinatesLet w = h(ρ, θ, φ ) be a functionover V in sphereical coordinates.We partion V by small incrementsof Δρ, Δθ, and Δφ.Spherical Coordinates
- 85. Triple Integrals in Cylindrical and Spherical CoordinatesLet w = h(ρ, θ, φ ) be a functionover V in sphereical coordinates.We partion V by small incrementsof Δρ, Δθ, and Δφ. A typicalΔV=ΔρxΔθxΔφ is a chunk of aspherical wedge.Spherical CoordinatesΔθXYZΔφΔρΔφ
- 86. Triple Integrals in Cylindrical and Spherical CoordinatesLet w = h(ρ, θ, φ ) be a functionover V in sphereical coordinates.We partion V by small incrementsof Δρ, Δθ, and Δφ. A typicalΔV=ΔρxΔθxΔφ is a chunk of aspherical wedge.Spherical CoordinatesΔθXYZΔφThe volume of ΔV depends on ρ-the distance tothe origin, and φ-the pitched angel.ΔρΔφ
- 87. Triple Integrals in Cylindrical and Spherical CoordinatesLet w = h(ρ, θ, φ ) be a functionover V in sphereical coordinates.We partion V by small incrementsof Δρ, Δθ, and Δφ. A typicalΔV=ΔρxΔθxΔφ is a chunk of aspherical wedge.Spherical CoordinatesΔθXYZΔφThe volume of ΔV depends on ρ-the distance tothe origin, and φ-the pitched angel.The larger the ρ is, the larger the volume of ΔV.The closer φ is to π/2, the larger the volume of ΔV is.ΔρΔφ
- 88. Triple Integrals in Cylindrical and Spherical CoordinatesLet w = h(ρ, θ, φ ) be a functionover V in sphereical coordinates.We partion V by small incrementsof Δρ, Δθ, and Δφ. A typicalΔV=ΔρxΔθxΔφ is a chunk of aspherical wedge.Spherical CoordinatesΔθXYZΔφThe volume of ΔV depends on ρ-the distance tothe origin, and φ-the pitched angel.The larger the ρ is, the larger the volume of ΔV.The closer φ is to π/2, the larger the volume of ΔV is.ΔρΔφIn fact, the volume of ΔV is (ρ*)2sin(φ*)ΔρΔθΔφ where(ρ*, θ*, φ*) is the center of ΔV.
- 89. Triple Integrals in Cylindrical and Spherical CoordinatesΔρ,Δθ,Δφ0Then∫ ∫∫vh(ρ, θ, φ )dV = lim Σ h(ρ*, θ*, φ*) ΔV
- 90. Triple Integrals in Cylindrical and Spherical CoordinatesΔρ,Δθ,Δφ0Then∫ ∫∫vh(ρ, θ, φ )dV = lim Σ h(ρ*, θ*, φ*)(ρ*)2sin(φ*) ΔρΔθΔφΔVΔρ,Δθ,Δφ0= lim Σ h(ρ*, θ*, φ*)
- 91. Triple Integrals in Cylindrical and Spherical CoordinatesΔρ,Δθ,Δφ0Then∫ ∫∫vh(ρ, θ, φ )dV = lim Σ h(ρ*, θ*, φ*)(ρ*)2sin(φ*) ΔρΔθΔφΔVΔρ,Δθ,Δφ0= lim Σ h(ρ*, θ*, φ*)ρ2sin(φ)=∫ ∫∫appropriate limitsfor integrationh(ρ, θ, φ ) dρ dθ dφ
- 92. Triple Integrals in Cylindrical and Spherical CoordinatesΔρ,Δθ,Δφ0Then∫ ∫∫vIts easier to express solids related to the spheres in spericalcoordinates:h(ρ, θ, φ )dV = lim Σ h(ρ*, θ*, φ*)(ρ*)2sin(φ*) ΔρΔθΔφΔVΔρ,Δθ,Δφ0= lim Σ h(ρ*, θ*, φ*)ρ2sin(φ)=∫ ∫∫appropriate limitsfor integrationh(ρ, θ, φ ) dρ dθ dφ
- 93. Triple Integrals in Cylindrical and Spherical CoordinatesΔρ,Δθ,Δφ0Then∫ ∫∫vIts easier to express solids related to the spheres in spericalcoordinates:h(ρ, θ, φ )dV = lim Σ h(ρ*, θ*, φ*)(ρ*)2sin(φ*) ΔρΔθΔφΔVΔρ,Δθ,Δφ0= lim Σ h(ρ*, θ*, φ*)ρ2sin(φ)=∫ ∫∫appropriate limitsfor integrationh(ρ, θ, φ ){0 < ρ < R; 0 < θ < 2π; 0 < φ < π }The sphere of radius R:Rdρ dθ dφ
- 94. Triple Integrals in Cylindrical and Spherical CoordinatesIf we varies ρ, we get the spherical shell:
- 95. Triple Integrals in Cylindrical and Spherical Coordinates{r < ρ < R; 0 < θ < 2π; 0 < φ < π }If we varies ρ, we get the spherical shell:Rr
- 96. Triple Integrals in Cylindrical and Spherical Coordinates{r < ρ < R; 0 < θ < 2π; 0 < φ < π }If we varies ρ, we get the spherical shell:RrIf we varies θ, we get the spherical wedge:
- 97. Triple Integrals in Cylindrical and Spherical Coordinates{r < ρ < R; 0 < θ < 2π; 0 < φ < π }If we varies ρ, we get the spherical shell:Rr{0 < ρ < R; A < θ < B; 0 < φ < π }If we varies θ, we get the spherical wedge:θ=Aθ=B
- 98. Triple Integrals in Cylindrical and Spherical CoordinatesIf we varies φ, we get the spherical cone:
- 99. Triple Integrals in Cylindrical and Spherical Coordinates{0 < ρ < R; 0 < θ < 2π; 0 < φ < C }If we varies φ, we get the spherical cone:Rφ=C
- 100. Triple Integrals in Cylindrical and Spherical Coordinates{0 < ρ < R; 0 < θ < 2π; 0 < φ < C }If we varies φ, we get the spherical cone:RIf we varies all tree variables, we get a chunk of thespherical wedge:φ=C
- 101. Triple Integrals in Cylindrical and Spherical Coordinates{0 < ρ < R; 0 < θ < 2π; 0 < φ < C }If we varies φ, we get the spherical cone:R{r < ρ < R; A < θ < B; C < φ < D }If we varies all tree variables, we get a chunk of thespherical wedge:θ=Aθ=Bφ=Cφ=Cρ=r φ=Dρ=R
- 102. Triple Integrals in Cylindrical and Spherical CoordinatesExample: Find∫ ∫∫vz dVwhere V is the solid bounded aboveby z = √1 – x2– y2and below byz = √3x2+ 3y2
- 103. Triple Integrals in Cylindrical and Spherical CoordinatesExample: Find∫ ∫∫vz dVwhere V is the solid bounded aboveby z = √1 – x2– y2and below byz = √3x2+ 3y2 1xy
- 104. Triple Integrals in Cylindrical and Spherical CoordinatesExample: Find∫ ∫∫vz dVwhere V is the solid bounded aboveby z = √1 – x2– y2and below byz = √3x2+ 3y2Put V in spherical coordinates.The pitched angle φ is π/6 becausethe border of the cone in theyz-plane is z = √3y.1xy
- 105. Triple Integrals in Cylindrical and Spherical CoordinatesExample: Find∫ ∫∫vz dVwhere V is the solid bounded aboveby z = √1 – x2– y2and below byz = √3x2+ 3y2Put V in spherical coordinates.The pitched angle φ is π/6 becausethe border of the cone in theyz-plane is z = √3y.1φ=π/6xy
- 106. Triple Integrals in Cylindrical and Spherical CoordinatesExample: Find∫ ∫∫vz dVwhere V is the solid bounded aboveby z = √1 – x2– y2and below byz = √3x2+ 3y2So V = {0 < ρ < 1;0 < θ < 2π ; 0 < φ < π/6}.The integrand z = ρcos(φ), hence the integral is1φ=π/6xyPut V in spherical coordinates.The pitched angle φ is π/6 becausethe border of the cone in theyz-plane is z = √3y.
- 107. Triple Integrals in Cylindrical and Spherical CoordinatesExample: Find∫ ∫∫vz dVwhere V is the solid bounded aboveby z = √1 – x2– y2and below byz = √3x2+ 3y2So V = {0 < ρ < 1;0 < θ < 2π ; 0 < φ < π/6}.The integrand z = ρcos(φ), hence the integral isρ2sin(φ)=∫ ∫∫ ρ=0ρ2cos(φ)π/6 2π 1θ=0φ=01φ=π/6xydρ dθ dφPut V in spherical coordinates.The pitched angle φ is π/6 becausethe border of the cone in theyz-plane is z = √3y.
- 108. Triple Integrals in Cylindrical and Spherical Coordinatesρ2sin(φ)∫ ∫∫ ρ=0ρ2cos(φ)π/6 2π 1θ=0φ=0dρ dθ dφ
- 109. Triple Integrals in Cylindrical and Spherical Coordinatesρ2sin(φ)∫ ∫∫ ρ=0ρ2cos(φ)π/6 2π 1θ=0φ=0sin(φ)= ∫ ∫ dθ dφcos(φ)π/6 2πθ=0φ=015dρ dθ dφ
- 110. Triple Integrals in Cylindrical and Spherical Coordinatesρ2sin(φ)∫ ∫∫ ρ=0ρ2cos(φ)π/6 2π 1θ=0φ=0sin(φ)= ∫ ∫ dθ dφcos(φ)π/6 2πθ=0φ=015sin(φ)= ∫ dφcos(φ)π/6φ=02π5dρ dθ dφ
- 111. Triple Integrals in Cylindrical and Spherical Coordinatesρ2sin(φ)∫ ∫∫ ρ=0ρ2cos(φ)π/6 2π 1θ=0φ=0sin(φ)= ∫ ∫ dθ dφcos(φ)π/6 2πθ=0φ=015sin(φ)= ∫ dφcos(φ)π/6φ=02π5sin2(φ) |=π/6φ=0π5=π20dρ dθ dφ
- 112. Triple Integrals in Cylindrical and Spherical CoordinatesA note on calculation of double and triple integrals:If the domnain of a double integral is a rectangle{a < x < b; c < y < d} and the integrand is of theform = f(x)g(y), then
- 113. Triple Integrals in Cylindrical and Spherical Coordinatesf(x) g(y)∫ ∫ dy dx =y=cx=aA note on calculation of double and triple integrals:If the domnain of a double integral is a rectangle{a < x < b; c < y < d} and the integrand is of theform = f(x)g(y), thenb dg(y)∫ ∫ dyy=cx=ab df(x)dx *
- 114. Triple Integrals in Cylindrical and Spherical Coordinatesf(x) g(y)∫ ∫ dy dx =y=cx=aA note on calculation of double and triple integrals:If the domnain of a double integral is a rectangle{a < x < b; c < y < d} and the integrand is of theform = f(x)g(y), thenb dg(y)∫ ∫ dyy=cx=ab df(x)dx *Similarly for triple integrals, weve:f(x) g(y) h(z)∫ ∫ dy dx dz =y=cx=ab dg(y)∫ ∫ dy *y=cx=ab df(x) dx*∫z=efh(z)∫ dzz=efExercise: Do the integral in the last example via this observation.

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