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AP Calculus 2010AB_4b Volume of Revolution Problem

AP Calculus 2010AB_4b Volume of Revolution Problem
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2016-06-26 22:27
FunctionSequenceSolid GeometryCalculusIntegral
In collections Volumes in AP Calculus Volumes in AP Calculus
This example demonstrates how to generate a 3D solid by revolving a planar region and compute its volume, serving as a dynamic visualization of part (b) of Problem 4 from the 2010 AP Calculus AB exam. Core mathematical concepts include definite integrals for volumes of revolution (disk/washer method), the relationship between function graphs and the axis of rotation, and the geometric construction of volume integrals. Construction highlights: using GeoGebra commands such as Sequence, Polygon, Rotate to generate cross-sections, Integral and Surface to compute and display the volume integral and the surface of revolution, RectangleSum and Rectangle objects to illustrate Riemann sum approximations, and Inner/Outer objects to represent the inner and outer radii in the washer method. Suitable for high school AP Calculus courses or introductory college calculus, this example helps students intuitively understand the derivation and calculation of volume formulas for solids of revolution.

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