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2010 AP Calculus AB Form B Problem 1b: Volume Problem

2010 AP Calculus AB Form B Problem 1b: Volume Problem
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2016-06-26 22:30
Solid GeometryCalculusIntegral
In collections Volumes in AP Calculus Volumes in AP Calculus
This example demonstrates the volume problem from the 2010 AP Calculus AB Form B, Problem 1b. Key mathematical concepts include integration and solid geometry. Using GeoGebra commands such as Sequence, Polygon, Rotate, Integral, Surface, etc., the process of calculating volume is constructed. Suitable for high school calculus courses and learning.

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 * ((8 - f(x))^(2) - (8 - g(x))^(2))), 0, 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, 8 - ((8 - g(((DK + p) * h))) * cos(β)), ((8 - g(((DK + p) * h))) * sin(β)), u, (DK * h), ((DK + 1) * h), β, 0, α) → Inner1
  9. Surface(u, 8 - ((8 - f(((DK + p) * h))) * cos(β)), ((8 - f(((DK + p) * h))) * sin(β)), u, (DK * h), ((DK + 1) * h), β, 0, α) → Outer1
  10. Surface(u, 8 + ((8 - g(u)) * cos(β)), ((8 - g(u)) * sin(β)), u, 0, 2, β, 0, (2 * π)) → Inner
  11. Surface(u, 8 - ((8 - f(u)) * cos(β)), ((8 - f(u)) * sin(β)), u, 0, 2, β, 0, (2 * π)) → Outer
  12. Surface((DK * h), 8 - ((8 - u) * cos(β)), ((8 - u) * sin(β)), u, f(((DK + p) * h)), g(((DK + p) * h)), β, 0, α) → SideLeft1
  13. Surface(((DK + 1) * h), 8 - ((8 - u) * cos(β)), ((8 - u) * sin(β)), u, f(((DK + p) * h)), g(((DK + p) * h)), β, 0, α) → SideRight1
  14. Sequence(Surface[u, 8 - ((8 - g(((k + p) * h))) * cos(β)), ((8 - g(((k + p) * h))) * sin(β)), u, (k * h), ((k + 1) * h), β, 0, α], k, 0, m - 1) → Inners
  15. Sequence(Surface[u, 8 - ((8 - f(((k + p) * h))) * cos(β)), ((8 - 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), 8 - ((8 - u) * cos(β)), ((8 - 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), 8 - ((8 - u) * cos(β)), ((8 - 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
  19. Segment((0, f(0)), (0, g(2), 0)) → SegL
  20. Segment((2, f(2)), (2, g(2), 0)) → SegR
  21. Rotate(SegL, α, a) → SegL'
  22. Rotate(SegR, α, a) → SegR'
  23. Surface(0, 8 - ((8 - u) * cos(β)), ((8 - u) * sin(β)), u, f(0), g(0), β, 0, (2 * π)) → SideLeft
  24. Surface(2, 8 - ((8 - u) * cos(β)), ((8 - u) * sin(β)), u, f(2), g(2), β, 0, (2 * π)) → SideRight

AI Drawing Prompt Reference

Draw a 3D shape formed by the projections of two functions f(x) and g(x) on the x-axis from x=0 to x=2. First, draw the projections of these functions on the x-axis, then rotate these projections around the x-axis to form two rotational surfaces. Next, calculate the volume between these two rotational surfaces. Finally, draw a series of rectangles that, when rotated around the x-axis, form the side surfaces.
The prompt above was compiled from the original drawing commands and is for reference only.