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Polar Area Calculation

Polar Area Calculation
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2020-01-11 05:46
CalculusIntegralLimit
This example demonstrates how to calculate area using polar coordinates. It involves core mathematical concepts such as integration and limits. Using GeoGebra commands like CurveCartesian and Integral, a polar area calculation model can be constructed. Suitable for high school and above students to learn methods of area calculation in calculus.

GeoGebra Commands & Parameters

  1. CurveCartesian(((g(t) * cos(t)), (g(t) * sin(t))), t, a_s, a) → b
  2. Sequence(a_s + (d / n * i), i, 0, n) → L_1
  3. Sequence(((1 - e) * Element[L_1, i]) + (e * Element[L_1, i + 1]), i, 1, n) → l3
  4. Textfield(f) → InputBox1
  5. Sequence(x = i, i, -30, 30) → l9
  6. Sequence(y = i, i, -30, 30) → l10
  7. Sequence(x^(2) + y^(2) = i^(2), i, 1, 20) → l11
  8. Textfield(a_s) → InputBox4
  9. Textfield(a_f) → InputBox5
  10. Integral((1 / 2 * (f(x))^(2)), a_s, a) → i
  11. Textfield(m) → InputBox6
  12. Textfield(a_s) → InputBox2_1
  13. Textfield(a_s) → InputBox4_1
  14. Textfield(a_s) → InputBox2_2
  15. Textfield(a_s) → InputBox4_2
  16. Textfield(a_s) → InputBox2_3
  17. Textfield(a_s) → InputBox4_3
  18. CurveCartesian(((f(t) * cos(t)), (f(t) * sin(t))), t, a_s, a_f) → c
  19. Sequence(b(Element[l3, i]), i, 1, n) → l1
  20. Sequence(b(Element[L_1, i]), i, 1, n + 1) → l2
  21. Sequence(Segment[A, Element[l1, i]], i, 1, n) → l4
  22. Sequence(If[Length[Element[l4, i]] > 0.0001, Length[Element[l4, i]], 0.0001], i, 1, n) → l5
  23. Sequence(Circle[A, Element[l5, i]], i, 1, n) → l6
  24. Sequence(Ray[A, Element[l2, i]], i, 1, n + 1) → l7
  25. Sequence(Intersect[Element[l6, i], Element[l7, i]], i, 1, n) → l8
  26. Sequence(Intersect[Element[l6, i], Element[l7, i + 1]], i, 1, n) → l12
  27. Sequence(CircleSector[A, Element[l8, i], Element[l12, i]], i, 1, n) → l13
  28. Sequence(Polygon[A, Element[l8, i], Element[l12, i]], i, 1, n) → l14
  29. Element(l14, 2) → t1
  30. Angle(Element[l8, 2], A, Element[l8, 3]) → α
  31. Segment(A, Element[l8, 3]) → h
  32. Element(l13, 2) → c_1
  33. Sequence(Area[Element[l13, i]], i, 1, n) → l15
  34. Sum(l15) → s_s
  35. Sequence(Area[Element[l14, i]], i, 1, n) → l16
  36. Sum(l16) → s_t

AI Drawing Prompt Reference

Draw a polar coordinate graph using the function f(t) = t^2. Calculate the area under this function in the interval [a_s, a_f] using integration. Draw a series of concentric circles and rays, calculate the area of each sector, and sum them up to get the total area.
The prompt above was compiled from the original drawing commands and is for reference only.