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Quadrature of the Parabola by John Golden

Quadrature of the Parabola by John Golden
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2014-05-13 12:28
FunctionCalculusGeometric Construction
This example demonstrates the construction and proof of calculating the area between a parabola and a line, known as a parabolic segment. It involves core mathematical concepts such as parabolas, integration, and geometric construction. The construction utilizes GeoGebra commands/objects like Mirror, FitPoly, Point, ParaTri, IntegralBetween, Sum, and Take. It is suitable for high school and above students to learn about calculus and geometric constructions.

GeoGebra Commands & Parameters

  1. Mirror(B, a) → B'
  2. FitPoly({B, A, B'}, 2) → f
  3. Point(f) → C
  4. Point(f) → D
  5. ParaTri(f, C, D) → E, poly1
  6. ParaTri(f, C, E) → F, poly2
  7. ParaTri(f, E, D) → G, poly3
  8. ParaTri(f, C, F) → H, poly4
  9. ParaTri(f, E, G) → I, poly5
  10. ParaTri(f, G, D) → J, poly6
  11. ParaTri(f, F, E) → K, poly7
  12. IntegralBetween(f, ((-(x(Line[C, D]) / y(Line[C, D]))) * x) - z(Line[C, D]) / y(Line[C, D]), x(C), x(D)) → b
  13. ParaTri(f, C, H) → L, poly8
  14. ParaTri(f, H, F) → M, poly9
  15. ParaTri(f, F, K) → N, poly10
  16. ParaTri(f, K, E) → O, poly11
  17. ParaTri(f, E, I) → P, poly12
  18. ParaTri(f, I, G) → Q, poly13
  19. ParaTri(f, G, J) → R, poly14
  20. ParaTri(f, J, D) → S, poly15
  21. Sum(list1, n) → c
  22. Take(list1, 1, n) → displaylist

AI Drawing Prompt Reference

Draw a parabolic segment between a parabola and a line. First, mirror point B to get B'. Then, fit a quadratic polynomial f through points B, A, and B'. Next, take points C and D on f, and construct triangles E, F, G, H, I, J, K, L, M, N, O, P, Q, R, and S. Finally, calculate the integral b between the parabola and the line, and display these triangles.
The prompt above was compiled from the original drawing commands and is for reference only.