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AP Calculus 2011 AB 3c Volume Problem GeoGebra Demo

AP Calculus 2011 AB 3c Volume Problem GeoGebra Demo
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2016-06-26 22:21
Solid GeometryCalculusIntegralGeometric ConstructionSlider Animation
In collections Volumes in AP Calculus Volumes in AP Calculus
This example demonstrates how to use GeoGebra to construct a model for the AP Calculus 2011 AB 3c volume problem, showcasing the calculation of irregular solid volumes through rotation and integration. Core math concepts include integration, rotational volumes, Pythagorean theorem, complex number operations, and polygon construction. Key GeoGebra commands and objects used are Sequence, Polygon, Rotate, Integral, Surface, and RectangleSum. Suitable for high school and above students to understand calculus volume computation, geometric transformations, and complex number applications, serving as a teaching aid or exercise supplement.

GeoGebra Commands & Parameters

  1. Sequence(Polygon[((k * h), f(((k + p) * h))), (((k + 1) * h), f(((k + p) * h))), (((k + 1) * h), g(((k + p) * h))), ((k * h), g(((k + p) * h)))], k, 0, m - 1) → Rectangles
  2. Polygon(((DK * h), f(((DK + p) * h))), (((DK + 1) * h), f(((DK + p) * h))), (((DK + 1) * h), g(((DK + p) * h))), ((DK * h), g(((DK + p) * h)))) → Rectangle1, b, c, d, e
  3. Rotate(Rectangle1, α, a) → Rectangle1'
  4. Integral((π * ((1 - f(x))^(2) - (1 - g(x))^(2))), 0, 1 / 2) → VolInteg
  5. Rotate(Rectangles, α, a) → Rectangles'
  6. Rotate(g_1, α, a) → i
  7. Rotate(f_1, α, a) → j
  8. Surface(u, 1 - ((1 - f(((DK + p) * h))) * cos(β)), ((1 - f(((DK + p) * h))) * sin(β)), u, (DK * h), ((DK + 1) * h), β, 0, α) → Inner1
  9. Surface(u, 1 - ((1 - g(((DK + p) * h))) * cos(β)), ((1 - g(((DK + p) * h))) * sin(β)), u, (DK * h), ((DK + 1) * h), β, 0, α) → Outer1
  10. Surface((DK * h), 1 - ((1 - u) * cos(β)), ((1 - u) * sin(β)), u, f(((DK + p) * h)), g(((DK + p) * h)), β, 0, α) → SideLeft1
  11. Surface(((DK + 1) * h), 1 - ((1 - u) * cos(β)), ((1 - u) * sin(β)), u, f(((DK + p) * h)), g(((DK + p) * h)), β, 0, α) → SideRight1
  12. Surface(u, 1 + ((1 - g(u)) * cos(β)), ((1 - g(u)) * sin(β)), u, 0, 1 / 2, β, 0, (2 * π)) → Inner
  13. Surface(u, 1 - ((1 - f(u)) * cos(β)), ((1 - f(u)) * sin(β)), u, 0, 1 / 2, β, 0, (2 * π)) → Outer
  14. Sequence(Surface[u, 1 - ((1 - g(((k + p) * h))) * cos(β)), ((1 - g(((k + p) * h))) * sin(β)), u, (k * h), ((k + 1) * h), β, 0, α], k, 0, m - 1) → Inners
  15. Sequence(Surface[u, 1 - ((1 - f(((k + p) * h))) * cos(β)), ((1 - f(((k + p) * h))) * sin(β)), u, (k * h), ((k + 1) * h), β, 0, α], k, 0, m - 1) → Outers
  16. Sequence(Surface[(k * h), 1 - ((1 - u) * cos(β)), ((1 - u) * sin(β)), u, f(((k + p) * h)), g(((k + p) * h)), β, 0, α], k, 0, m - 1) → SidesLeft
  17. Sequence(Surface[((k + 1) * h), 1 - ((1 - u) * cos(β)), ((1 - u) * sin(β)), u, f(((k + p) * h)), g(((k + p) * h)), β, 0, α], k, 0, m - 1) → SidesRight
  18. RectangleSum(A, 0, (m * h), m, p) → WS