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2007 AP AB Form B Question 1c: Volume of Revolution

2007 AP AB Form B Question 1c: Volume of Revolution
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2016-06-26 23:43
Solid GeometryCalculusIntegral
In collections Volumes in AP Calculus Volumes in AP Calculus
This example illustrates the volume of revolution problem from the 2007 AP AB Form B Question 1c. Key mathematical concepts include integration and solid geometry. Using GeoGebra commands such as Intersect, Sequence, Polygon, etc., the volume of the revolution body can be constructed. Suitable for high school math education and learning.

GeoGebra Commands & Parameters

  1. Intersect(g, f, 0, 2) → A, B
  2. Sequence(Cylinder[(a + (k * h), 1, 0), (a + ((k + 1) * h), 1, 0), 1 + f(a + ((k + p) * h))], k, 0, m - 1) → CO
  3. Sequence(Cylinder[(a + (k * h), 1, 0), (a + ((k + 1) * h), 1, 0), 1 + g(a + ((k + p) * h))], k, 0, m - 1) → CI
  4. Sequence(Polygon[(a + (k * h), f(a + ((k + p) * h))), (a + ((k + 1) * h), f(a + ((k + p) * h))), (a + ((k + 1) * h), g(a + ((k + p) * h))), (a + (k * h), g(a + ((k + p) * h)))], k, 0, m - 1) → Rectangles
  5. Sequence(If[Defined[Volume[Element[CO, k]]], Volume[Element[CO, k]], 0], k, 1, m) → VCO
  6. Sequence(If[Defined[Volume[Element[CI, k]]], Volume[Element[CI, k]], 0], k, 1, m) → VCI
  7. Sum(VCO - VCI) → VC
  8. Polygon((a + (DK * h), g(a + ((DK + p) * h))), (a + ((DK + 1) * h), g(a + ((DK + p) * h))), (a + ((DK + 1) * h), f(a + ((DK + p) * h))), (a + (DK * h), f(a + ((DK + p) * h)))) → Rectangle1, c, d, e, k
  9. Rotate(Rectangle1, (-α), line) → Rectangle1'
  10. Integral((π * ((g(x) - 1)^(2) - (f(x) - 1)^(2))), a, b) → VolInteg
  11. Rotate(f_1, (-α), line) → i
  12. Rotate(g_1, (-α), line) → j
  13. Rotate(Rectangles, (-α), line) → Rectangles'
  14. Surface(u, 1 + ((f(u) - 1) * cos(β)), ((f(u) - 1) * sin(β)), u, a, b, β, 0, (2 * π)) → Inner
  15. Surface(a + (DK * h), 1 + ((u - 1) * cos(β)), ((u - 1) * sin(β)), u, f(a + ((DK + p) * h)), g(a + ((DK + p) * h)), β, 0, α) → SideLeft1
  16. Surface(a + ((DK + 1) * h), 1 + ((u - 1) * cos(β)), ((u - 1) * sin(β)), u, f(a + ((DK + p) * h)), g(a + ((DK + p) * h)), β, 0, α) → SideRight1
  17. Surface(u, 1 + ((g(u) - 1) * cos(β)), ((g(u) - 1) * sin(β)), u, a, b, β, 0, (2 * π)) → Outer
  18. Surface(u, 1 + ((f(a + ((DK + p) * h)) - 1) * cos(β)), ((f(a + ((DK + p) * h)) - 1) * sin(β)), u, a + (DK * h), a + ((DK + 1) * h), β, 0, α) → Inner1
  19. Surface(u, 1 + ((g(a + ((DK + p) * h)) - 1) * cos(β)), ((g(a + ((DK + p) * h)) - 1) * sin(β)), u, a + (DK * h), a + ((DK + 1) * h), β, 0, α) → Outer1
  20. Sequence(Surface[u, 1 + ((g(a + ((k + p) * h)) - 1) * cos(β)), ((g(a + ((k + p) * h)) - 1) * sin(β)), u, a + (k * h), a + ((k + 1) * h), β, 0, α], k, 0, m - 1) → Inners
  21. Sequence(Surface[u, 1 + ((f(a + ((k + p) * h)) - 1) * cos(β)), ((f(a + ((k + p) * h)) - 1) * sin(β)), u, a + (k * h), a + ((k + 1) * h), β, 0, α], k, 0, m - 1) → Outers
  22. Sequence(Surface[a + (k * h), 1 + ((u - 1) * cos(β)), ((u - 1) * sin(β)), u, f(a + ((k + p) * h)), g(a + ((k + p) * h)), β, 0, α], k, 0, m - 1) → SidesLeft
  23. Sequence(Surface[a + ((k + 1) * h), 1 + ((u - 1) * cos(β)), ((u - 1) * sin(β)), u, f(a + ((k + p) * h)), g(a + ((k + p) * h)), β, 0, α], k, 0, m - 1) → SidesRight
  24. RectangleSum((π * ((ℯ^((2 * x) - x^(2)) - 1)^(2) - 1)), a, a + (m * h), m, p) → VCyls

AI Drawing Prompt Reference

Illustrate the volume of a solid of revolution, showcasing integral and solid geometry concepts. Start by plotting the intersection points of two curves with the x-axis, then construct the rotating body using stacked cylinders and rectangles. Display the volume calculation with integral expressions, showing how the solid is built up from small volumes.
The prompt above was compiled from the original drawing commands and is for reference only.