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Volume with Semicircular Cross Sections

Volume with Semicircular Cross Sections
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2016-03-18 03:09
FunctionSolid GeometryCalculusIntegral
In collections AP Calculus volume problems
This example demonstrates the calculation of volume with semicircular cross sections and a known base, involving key calculus concepts like integration, function graphing, and geometric construction. It utilizes GeoGebra commands such as Segment, Integral, Semicircle, and Sequence for construction, making it suitable for high school and above students learning calculus and geometric modeling.

GeoGebra Commands & Parameters

  1. Segment(A, B) → e
  2. Integral((π / 2 * r(x)^(2)), -2, 2) → Integ
  3. If(togDF ≟ 1, "Show Instructions", "Show volume integral") → ToggleDirectionVolumeIntegral
  4. If(togS ≟ 1, "Hide surface", "Show surface") → ShowSurface
  5. Plane((0, 0, 0), (0, 1, 1), (0, 2, 0)) → a
  6. Surface(u, (v * f(u)) + ((1 - v) * g(u)), 0, u, -2, 2, v, 0, 1) → Base
  7. Semicircle(A, B, a) → SC
  8. Sequence(Surface[u, c(-2 + ((k + p) * h)) + (r(-2 + ((k + p) * h)) * cos(β)), (r(-2 + ((k + p) * h)) * sin(β)), u, -2 + (h * k), -2 + ((k + 1) * h), β, 0, π], k, 0, m - 1) → SlicesTop
  9. Sequence(Semicircle[(-2 + (k * h), g(-2 + ((k + p) * h)), 0), (-2 + (k * h), f(-1 + ((k + p) * h)), 0), a], k, 0, m - 1) → SlicesLeft
  10. Sequence(Semicircle[(-2 + ((k + 1) * h), g(-2 + ((k + p) * h)), 0), (-2 + ((k + 1) * h), f(-1 + ((k + p) * h)), 0), a], k, 0, m - 1) → SlicesRight
  11. Sum(Sequence[((h * π / 8) * (g(-2 + ((k + p) * h)) - f(-2 + ((k + p) * h)))^(2)), k, 0, m - 1]) → VolSlices
  12. Sequence(Polygon[(-2 + (k * h), f(-2 + ((k + p) * h))), (-2 + (k * h), g(-2 + ((k + p) * h))), (-2 + ((k + 1) * h), g(-2 + ((k + p) * h))), (-2 + ((k + 1) * h), f(-2 + ((k + p) * h)))], k, 0, m - 1) → SlicesBase
  13. Surface(u, c(u) + (r(u) * cos(β)), (r(u) * sin(β)), u, 0, 2, β, 0, π) → TopRight
  14. Surface(u, c(u) + (r(u) * cos(β)), (r(u) * sin(β)), u, -2, 0, β, 0, π) → TopLeft