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2006 AP AB Form B Problem 1b: Volume Problem

2006 AP AB Form B Problem 1b: Volume Problem
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2016-06-26 23:59
FunctionCalculusIntegral
In collections Volumes in AP Calculus Volumes in AP Calculus
This example demonstrates the volume problem from 2006 AP AB Form B. Core mathematical concepts include integration and the volume of a solid of revolution. Using GeoGebra commands such as Intersect, Polygon, and Integral, the volume of the solid can be constructed. Suitable for high school math education and students learning calculus.

GeoGebra Commands & Parameters

  1. Intersect(f, xAxis, (-1.3731216646760573, 0)) → B
  2. Sequence(Polygon[(aa + (k * h), f(aa + ((k + p) * h))), (aa + ((k + 1) * h), f(aa + ((k + p) * h))), (aa + ((k + 1) * h), g(aa + ((k + p) * h))), (aa + (k * h), g(aa + ((k + p) * h)))], k, 0, m - 1) → Rectangles, {0.11989168777938175, 0.32799428683256315, 0.4911547291915701, 0.60700600002159}
  3. Polygon((aa + (DK * h), f(aa + ((DK + p) * h))), (aa + ((DK + 1) * h), f(aa + ((DK + p) * h))), (aa + ((DK + 1) * h), g(aa + ((DK + p) * h))), (aa + (DK * h), g(aa + ((DK + p) * h)))) → Rectangle1, b, c, d, e
  4. Rotate(Rectangle1, (-α), a) → Rectangle1'
  5. Integral((π * ((2 + f(x))^(2) - 4)), aa, 0) → VolInteg
  6. RectangleSum(A, aa, aa + (m * h), m, p) → WS
  7. Rotate(Rectangles, (-α), a) → Rectangles'
  8. Sequence(Surface[aa + ((k + u) * h), -2 + (2 * cos(β)), (2 * sin(β)), u, 0, 1, β, 0, α], k, 0, m - 1) → Inners, {(-1.3731216646760573 + ((u) * 0.22885361077934288), -2 + (2 * cos(β)), (2 * sin(β))), (-1.3731216646760573 + ((1 + u) * 0.22885361077934288), -2 + (2 * cos(β)), (2 * sin(β))), (-1.3731216646760573 + ((2 + u) * 0.22885361077934288), -2 + (2 * cos(β)), (2 * sin(β))), (-1.3731216646760573 + ((3 + u) * 0.22885361077934288), -2 + (2 * cos(β)), (2 * sin(β)))}
  9. Sequence(Surface[aa + ((k + u) * h), -2 + ((2 + f(aa + ((k + p) * h))) * cos(β)), ((2 + f(aa + ((k + p) * h))) * sin(β)), u, 0, 1, β, 0, α], k, 0, m - 1) → Outers, {(-1.3731216646760573 + ((u) * 0.22885361077934288), -2 + ((2 + f(-1.3731216646760573 + ((0.5) * 0.22885361077934288))) * cos(β)), ((2 + f(-1.3731216646760573 + ((0.5) * 0.22885361077934288))) * sin(β))), (-1.3731216646760573 + ((1 + u) * 0.22885361077934288), -2 + ((2 + f(-1.3731216646760573 + ((1 + 0.5) * 0.22885361077934288))) * cos(β)), ((2 + f(-1.3731216646760573 + ((1 + 0.5) * 0.22885361077934288))) * sin(β))), (-1.3731216646760573 + ((2 + u) * 0.22885361077934288), -2 + ((2 + f(-1.3731216646760573 + ((2 + 0.5) * 0.22885361077934288))) * cos(β)), ((2 + f(-1.3731216646760573 + ((2 + 0.5) * 0.22885361077934288))) * sin(β))), (-1.3731216646760573 + ((3 + u) * 0.22885361077934288), -2 + ((2 + f(-1.3731216646760573 + ((3 + 0.5) * 0.22885361077934288))) * cos(β)), ((2 + f(-1.3731216646760573 + ((3 + 0.5) * 0.22885361077934288))) * sin(β)))}
  10. Rotate(g_1, (-α), a) → i
  11. Rotate(f_1, (-α), a) → j
  12. Surface(u, -2 + ((2 + g(u)) * cos(β)), ((2 + g(u)) * sin(β)), u, aa, 0, β, 0, (2 * π)) → Inner
  13. Surface(aa + ((k + u) * h), -2 + ((2 + g(aa + ((DK + p) * h))) * cos(β)), ((2 + g(aa + ((DK + p) * h))) * sin(β)), u, (DK * h), ((DK + 1) * h), β, 0, α) → Inner1
  14. Surface(aa + ((k + u) * h), -2 + ((2 + f(aa + ((DK + p) * h))) * cos(β)), ((2 + f(aa + ((DK + p) * h))) * sin(β)), u, (DK * h), ((DK + 1) * h), β, 0, α) → Outer1
  15. Surface(aa + (DK * h), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β)), u, f(aa + ((DK + p) * h)), g(aa + ((DK + p) * h)), β, 0, α) → SideLeft1
  16. Surface(aa + ((DK + 1) * h), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β)), u, f(aa + ((DK + p) * h)), g(aa + ((DK + p) * h)), β, 0, α) → SideRight1
  17. Surface(u, -2 + ((2 + f(u)) * cos(β)), ((2 + f(u)) * sin(β)), u, aa, 0, β, 0, (2 * π)) → Outer
  18. Sequence(Surface[aa + (k * h), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β)), u, 0, f(aa + ((k + p) * h)), β, 0, α], k, 0, m - 1) → SidesLeft, {(-1.3731216646760573 + (0 * 0.22885361077934288), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β))), (-1.3731216646760573 + (1 * 0.22885361077934288), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β))), (-1.3731216646760573 + (2 * 0.22885361077934288), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β))), (-1.3731216646760573 + (3 * 0.22885361077934288), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β)))}
  19. Sequence(Surface[aa + ((k + 1) * h), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β)), u, 0, f(aa + ((k + p) * h)), β, 0, α], k, 0, m - 1) → SidesRight, {(-1.3731216646760573 + ((1) * 0.22885361077934288), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β))), (-1.3731216646760573 + ((1 + 1) * 0.22885361077934288), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β))), (-1.3731216646760573 + ((2 + 1) * 0.22885361077934288), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β))), (-1.3731216646760573 + ((3 + 1) * 0.22885361077934288), -2 + ((2 + u) * cos(β)), ((2 + u) * sin(β)))}
  20. Segment((aa, 0), (0, 0)) → l
  21. Segment((0, 0), (0, f(0))) → q
  22. Rotate(q, (-α), a) → q'