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Finding the Area Enclosed by a Polar Curve

Finding the Area Enclosed by a Polar Curve
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2020-08-22 19:27
Coordinate GeometryCalculusIntegralGeometric Construction
This example demonstrates how to calculate the area enclosed by a polar curve using polar coordinates and integration. It involves constructing a polar curve, computing the area of a circular sector, and applying the integral formula for polar area. The GeoGebra commands include CurveCartesian for curve plotting, CircleSector and Segment for geometric construction, Integral for area calculation, and Midpoint/Rotate for geometric manipulation. Suitable for high school and early university calculus students learning polar coordinate geometry.

GeoGebra Commands & Parameters

  1. CurveCartesian(((1 + cos(t)) * cos(t)), ((1 + cos(t)) * sin(t)), t, (-π), π) → a
  2. CurveCartesian(((1 + cos(t)) * cos(t)), ((1 + cos(t)) * sin(t)), t, 0, π) → c
  3. FractionText(d) → text1
  4. Rotate(Q, α, O) → Q'
  5. Angle(Q, O, Q') → β
  6. CircleSector(O, Q, Q') → δA
  7. FractionText(deltatheta) → text3
  8. Segment(O, Q') → f
  9. Intersect(f, a) → B, P, F, G
  10. Rotate(P, (-β), O) → P'
  11. Angle(P', O, P) → γ
  12. CircleSector(O, P', P) → e
  13. Midpoint(O, Q) → A
  14. Midpoint(O, P) → C
  15. Midpoint(P', Q) → D
  16. Angle(E, O, Q) → δ
  17. Midpoint(A, C) → H
  18. CircleSector(O, P', P) → g
  19. Integral((1 + cos(x))^(2), theta, theta + deltatheta) → Area
  20. Sequence((1 + cos(θ); θ), θ, 0, π, π / n) → l1
  21. Ray(O, l1(1)) → m
  22. Ray(O, l1(2)) → p
  23. Ray(O, l1(3)) → q
  24. Ray(O, l1(4)) → r
  25. Ray(O, l1(5)) → s
  26. Ray(O, l1(6)) → t
  27. Ray(O, l1(7)) → f_1
  28. Ray(O, l1(8)) → g_1
  29. Ray(O, l1(9)) → h_1
  30. Ray(O, l1(10)) → i_1
  31. Ray(O, l1(11)) → j_1
  32. Ray(O, l1(12)) → k_1
  33. Ray(O, l1(13)) → l_1
  34. Ray(O, l1(14)) → m_1
  35. Ray(O, l1(15)) → n_1
  36. Ray(O, l1(16)) → p_1
  37. Ray(O, l1(17)) → q_1
  38. Ray(O, l1(18)) → r_1
  39. Ray(O, l1(19)) → s_1
  40. Ray(O, l1(20)) → t_1
  41. Sequence((1 + cos(θ); θ + π / n), θ, 0, π, π / n) → l2
  42. CircleSector(O, l1(1), l2(1)) → Out_1
  43. CircleSector(O, l1(2), l2(2)) → Out_2
  44. CircleSector(O, l1(3), l2(3)) → Out_3
  45. CircleSector(O, l1(4), l2(4)) → Out_4
  46. CircleSector(O, l1(5), l2(5)) → Out_5
  47. CircleSector(O, l1(6), l2(6)) → Out_6
  48. CircleSector(O, l1(7), l2(7)) → Out_7
  49. CircleSector(O, l1(8), l2(8)) → Out_8
  50. CircleSector(O, l1(9), l2(9)) → Out_9
  51. CircleSector(O, l1(10), l2(10)) → Out_{10}
  52. CircleSector(O, l1(11), l2(11)) → Out_{11}
  53. CircleSector(O, l1(12), l2(12)) → Out_{12}
  54. CircleSector(O, l1(13), l2(13)) → Out_{13}
  55. CircleSector(O, l1(14), l2(14)) → Out_{14}
  56. CircleSector(O, l1(15), l2(15)) → Out_{15}
  57. CircleSector(O, l1(16), l2(16)) → Out_{16}
  58. CircleSector(O, l1(17), l2(17)) → Out_{17}
  59. CircleSector(O, l1(18), l2(18)) → Out_{18}
  60. CircleSector(O, l1(19), l2(19)) → Out_{19}
  61. CircleSector(O, l1(20), l2(20)) → Out_{20}
  62. Sequence((1 + cos(θ + π / n); θ), θ, 0, π, π / n) → l3
  63. CircleSector(O, l3(1), l2(1)) → In_1
  64. CircleSector(O, l3(2), l2(2)) → In_{2}
  65. CircleSector(O, l3(3), l2(3)) → In_4
  66. CircleSector(O, l3(4), l2(4)) → In_{3}
  67. CircleSector(O, l3(5), l2(5)) → In_5
  68. CircleSector(O, l3(6), l2(6)) → In_{6}
  69. CircleSector(O, l3(7), l2(7)) → In_7
  70. CircleSector(O, l3(8), l2(8)) → In_8
  71. CircleSector(O, l3(9), l2(9)) → In_9
  72. CircleSector(O, l3(10), l2(10)) → In_{10}
  73. CircleSector(O, l3(11), l2(11)) → In_{11}
  74. CircleSector(O, l3(12), l2(12)) → In_{12}
  75. CircleSector(O, l3(13), l2(13)) → In_{13}
  76. CircleSector(O, l3(14), l2(14)) → In_{14}
  77. CircleSector(O, l3(15), l2(15)) → In_{15}
  78. CircleSector(O, l3(16), l2(16)) → In_{16}
  79. CircleSector(O, l3(17), l2(17)) → In_{17}
  80. CircleSector(O, l3(18), l2(18)) → In_{18}
  81. CircleSector(O, l3(19), l2(19)) → In_{19}
  82. CircleSector(O, l3(20), l2(20)) → In_{20}
  83. CircleSector(O, l3(1), l2(1)) → Outl_1
  84. CircleSector(O, l3(2), l2(2)) → Outl_2
  85. CircleSector(O, l3(3), l2(3)) → Outl_1_3
  86. CircleSector(O, l3(4), l2(4)) → Outl_4
  87. CircleSector(O, l3(5), l2(5)) → Outl_5
  88. CircleSector(O, l3(6), l2(6)) → Outl_6
  89. CircleSector(O, l3(7), l2(7)) → Outl_7
  90. CircleSector(O, l3(8), l2(8)) → Outl_3
  91. CircleSector(O, l3(8), l2(8)) → Outl_8
  92. CircleSector(O, l3(9), l2(9)) → Outl_9
  93. CircleSector(O, l3(10), l2(10)) → Outl_{10}
  94. CircleSector(O, l3(11), l2(11)) → Outl_{11}
  95. CircleSector(O, l3(12), l2(12)) → Outl_{12}
  96. CircleSector(O, l3(13), l2(13)) → Outl_{13}
  97. CircleSector(O, l3(14), l2(14)) → Outl_{14}
  98. CircleSector(O, l3(15), l2(15)) → Outl_{15}
  99. CircleSector(O, l3(16), l2(16)) → Outl_{16}
  100. CircleSector(O, l3(17), l2(17)) → Outl_{17}
  101. CircleSector(O, l3(18), l2(18)) → Outl_{18}
  102. CircleSector(O, l3(19), l2(19)) → Outl_{19}

AI Drawing Prompt Reference

On a coordinate plane, plot a polar-type parametric curve using two parameter ranges: draw one curve for angle parameter from −π to π and another for angle parameter from 0 to π, keeping them as the two reference curves. Select curve points that will act as endpoints of sector constructions, then use rotation about the origin to create a corresponding endpoint related by a measured angle. Construct sector arcs at the origin bounded by the rotated and original endpoints, and include the straight segment(s) connecting the origin to the endpoints. Find intersection points between the constructed sector boundary/segment and the drawn curve(s), then rotate one of the intersection-derived points about the origin by the indicated angle direction to obtain the second endpoint. Add the corresponding sector (arc) around the origin for the rotated configuration, preparing the visual for using the parameter increment as a reference for the area calculation over the interval.
The prompt above was compiled from the original drawing commands and is for reference only.