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Semicircle Cross Sections

Semicircle Cross Sections
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2016-08-12 07:14
CircleIntegralGeometric Construction
In collections BC Calculus
This example demonstrates the geometric construction of semicircle cross sections. Key mathematical concepts include circles and similar triangles. Using GeoGebra commands such as Semicircle, Sequence, and Integral, the formation process of semicircle cross sections is dynamically illustrated. Suitable for high school geometry and calculus courses.

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