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# 27 triple integrals in spherical and cylindrical coordinates

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### 27 triple integrals in spherical and cylindrical coordinates

1. 1. Cylindrical and Spherical Coordinates
2. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 12. Cylindrical and Spherical CoordinatesThe constant equationsr = k describes thecylinder of radius k, thusthe name "cylindricalcoordinate".Example: Sketch r = 2
13. 13. Cylindrical and Spherical CoordinatesThe constant equationsr = k describes thecylinder of radius k, thusthe name "cylindricalcoordinate".Example: Sketch r = 22
14. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 26. Cylindrical and Spherical CoordinatesExample: Plot the point (3, 2π/3, π/4) in cylindricalcoordinate. Convert it to the other systems.
27. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 38. Cylindrical and Spherical CoordinatesThe constant equationsρ = k describe the sphereof radius k, thus the name"spherical coordinate".Example: Sketch ρ = 2
39. 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. 40. Cylindrical and Spherical CoordinatesExample: Convert the equation z = √k2– x2– y2intothe polar equation and the spherical equation.
41. 41. Cylindrical and Spherical CoordinatesExample: Convert the equation z = √k2– x2– y2intothe polar equation and the spherical equation.k
42. 42. Cylindrical and Spherical CoordinatesExample: Convert the equation z = √k2– x2– y2intothe polar equation and the spherical equation.Since r = x2+ y2,k
43. 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. 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. 45. Cylindrical and Spherical CoordinatesExample: Convert the equation z = x2+ y2into thepolar equation and the spherical equation.
46. 46. Cylindrical and Spherical CoordinatesExample: Convert the equation z = x2+ y2into thepolar equation and the spherical equation.
47. 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. 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. 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. 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. 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. 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. 53. Triple Integrals in Cylindrical and Spherical Coordinates
54. 54. Triple Integrals in Cylindrical and Spherical CoordinatesA solids V may be represented easier in cylindricalor sphereical format than in rectangularcoordinates.
55. 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. 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. 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. 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. 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. 60. Triple Integrals in Cylindrical and Spherical CoordinatesIn particular, let r*be thedistance to the center ofΔrxΔθ,ΔθXYZΔrΔzr*
61. 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. 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. 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,Δz0then h(r, θ, z) dV∫ ∫∫vLet (ri, θi, zi) is the center point of each wedge in the partition,(ri, θi, zi)
64. 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,Δz0then 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. 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,Δz0then 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. 66. Triple Integrals in Cylindrical and Spherical CoordinatesXYZπ/6π/6357VExample: Write V usingcylindrical coordinates.
67. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 78. Triple Integrals in Cylindrical and Spherical Coordinates1* r dz dr dθ∫ ∫∫2πθ=0 r=03z=1√25 – r2
79. 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. 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. 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. 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. 83. Triple Integrals in Cylindrical and Spherical CoordinatesLet w = h(ρ, θ, φ ) be a functionover V in sphereical coordinates.Spherical Coordinates
84. 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. 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. 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. 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. 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. 89. Triple Integrals in Cylindrical and Spherical CoordinatesΔρ,Δθ,Δφ0Then∫ ∫∫vh(ρ, θ, φ )dV = lim Σ h(ρ*, θ*, φ*) ΔV
90. 90. Triple Integrals in Cylindrical and Spherical CoordinatesΔρ,Δθ,Δφ0Then∫ ∫∫vh(ρ, θ, φ )dV = lim Σ h(ρ*, θ*, φ*)(ρ*)2sin(φ*) ΔρΔθΔφΔVΔρ,Δθ,Δφ0= lim Σ h(ρ*, θ*, φ*)
91. 91. Triple Integrals in Cylindrical and Spherical CoordinatesΔρ,Δθ,Δφ0Then∫ ∫∫vh(ρ, θ, φ )dV = lim Σ h(ρ*, θ*, φ*)(ρ*)2sin(φ*) ΔρΔθΔφΔVΔρ,Δθ,Δφ0= lim Σ h(ρ*, θ*, φ*)ρ2sin(φ)=∫ ∫∫appropriate limitsfor integrationh(ρ, θ, φ ) dρ dθ dφ
92. 92. Triple Integrals in Cylindrical and Spherical CoordinatesΔρ,Δθ,Δφ0Then∫ ∫∫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,Δφ0= lim Σ h(ρ*, θ*, φ*)ρ2sin(φ)=∫ ∫∫appropriate limitsfor integrationh(ρ, θ, φ ) dρ dθ dφ
93. 93. Triple Integrals in Cylindrical and Spherical CoordinatesΔρ,Δθ,Δφ0Then∫ ∫∫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,Δφ0= 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. 94. Triple Integrals in Cylindrical and Spherical CoordinatesIf we varies ρ, we get the spherical shell:
95. 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. 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. 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. 98. Triple Integrals in Cylindrical and Spherical CoordinatesIf we varies φ, we get the spherical cone:
99. 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. 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. 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. 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. 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. 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. 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. 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. 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. 108. Triple Integrals in Cylindrical and Spherical Coordinatesρ2sin(φ)∫ ∫∫ ρ=0ρ2cos(φ)π/6 2π 1θ=0φ=0dρ dθ dφ
109. 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. 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. 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. 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. 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. 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.