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Area Between 2 Polar Graphs

Area Between 2 Polar Graphs
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2016-04-20 03:00
FunctionTrigonometryCoordinate GeometryCalculusIntegral
This example demonstrates how to calculate the area bounded by two polar functions. It involves core mathematical concepts like polar coordinate functions, integral calculus, and area computation. The construction utilizes GeoGebra commands such as Curve, Sequence, Textfield, FractionText, and Sum, along with objects like r, a, tmin, tmax, n, b, lists 1-6, InputBoxes 1-5, and various text and variable objects. It is suitable for high school and above students learning about the graphs and properties of polar functions and the application of integrals in geometry.

GeoGebra Commands & Parameters

  1. CurveCartesian((r(t) * cos(t)), (r(t) * sin(t)), t, 0, 6.283185307179586) → a
  2. Sequence(a(i), i, tmin, tmax, b) → list1
  3. Sequence(Segment[(0, 0), Element[list1, i]], i, 0, n) → list2
  4. Sequence(Polygon[(0, 0), Element[list1, i], Element[list1, i + 1]], i, 1, n) → list3
  5. Textfield(tmin) → InputBox1
  6. Textfield(tmax) → InputBox2
  7. Textfield(n) → InputBox3
  8. Textfield(r) → InputBox4
  9. Sequence(Segment[(0, 0), Element[list1, i]], i, 1, n) → list4
  10. FractionText(tmin / π) → text2
  11. FractionText(tmax / π) → text3
  12. Sum(list3, n) → c
  13. FractionText(d) → text5
  14. CurveCartesian((s(t) * cos(t)), (s(t) * sin(t)), t, 0, 6.283185307179586) → e
  15. Textfield(s) → InputBox5
  16. Sequence(e(i), i, tmin, tmax, b) → list5
  17. Sequence(Polygon[(0, 0), Element[list5, i], Element[list5, i + 1]], i, 1, n) → list6
  18. Sum(list6, n) → k
  19. Sequence(tmin + (b * i), i, 0, n, 1) → ENDS
  20. Sequence((0.5 * (Element[ENDS, i] + Element[ENDS, i + 1])), i, 1, n, 1) → REALCENTERS
  21. Sequence(a(Element[REALCENTERS, i]), i, 1, n, 1) → BLUECENTERS
  22. Sequence(Rotate[Element[BLUECENTERS, i], halftheta, A], i, 1, n, 1) → BLUEUPPERS
  23. Sequence(Rotate[Element[BLUECENTERS, i], (-halftheta), A], i, 1, n, 1) → BLUELOWERS
  24. Sequence(Sector[Circle[A, Element[BLUECENTERS, i]], Element[BLUELOWERS, i], Element[BLUEUPPERS, i]], i, 1, n, 1) → GREATERSECTORS
  25. Sequence(e(Element[REALCENTERS, i]), i, 1, n, 1) → GREENCENTERS
  26. Sequence(Rotate[Element[GREENCENTERS, i], halftheta, A], i, 1, n, 1) → GREENUPPERS
  27. Sequence(Rotate[Element[GREENCENTERS, i], (-halftheta), A], i, 1, n, 1) → GREENLOWERS
  28. Sequence(Sector[Circle[A, Element[GREENCENTERS, i]], Element[GREENLOWERS, i], Element[GREENUPPERS, i]], i, 1, n, 1) → LESSERSECTORS

AI Drawing Prompt Reference

Draw the area bounded by two polar functions, first plot the curve of the first function, then approximate the area it encloses by constructing multiple sectors, next plot the curve of the second function and construct its sectors, finally subtract the two areas to get the result, demonstrating the geometric meaning of polar coordinate functions, integral calculus, and area computation.
The prompt above was compiled from the original drawing commands and is for reference only.