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Volume Calculation with Right Triangular Cross-Sections Between Two Quadrilatera

Volume Calculation with Right Triangular Cross-Sections Between Two Quadrilatera
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2015-07-02 12:42
TrigonometrySolid GeometryCalculusIntegralGeometric Construction
In collections intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve Volumes in Calculus Volumes in Calculus
This example demonstrates how to calculate the volume formed by right triangular cross-sections between two parallelograms. Key math concepts include integration, solid geometry, and properties of right triangles. GeoGebra commands and objects used in construction are Polygon, Angle, Locus, Integral, If, Surface, and Sequence. Suitable for high school and above students to learn volume calculation methods and the application of integration in geometry.

GeoGebra Commands & Parameters

  1. Polygon(B, A, C) → Triangle, c_1, b_1, a_1
  2. Angle(B, A, C) → α
  3. Locus(C, a) → loc1
  4. Integral((4 + (2 * x) - (2 * x^(2)))^(2), -1, 2) → Integ
  5. If(togDF ≟ 1, "Show Instructions", "Show volume integral") → ToggleDirectionVolumeIntegral
  6. If(togS ≟ 1, "Hide surface", "Show surface") → ShowSurface
  7. Surface(u, (v * f(u)) + ((1 - v) * g(u)), 0, u, -1, 2, v, 0, 1) → Base
  8. Surface(u, g(u), ((2 * v) * (g(u) - f(u))), v, 0, 1, u, -1, 2) → SurfaceBack
  9. Surface(u, (v * f(u)) + ((1 - v) * g(u)), (2 * ((v * f(u)) + ((1 - v) * g(u)) - f(u))), v, 0, 1, u, -1, 2) → SurfaceFront
  10. Sequence(Polygon[(-1 + (k * h), f(-1 + ((k + p) * h))), (-1 + (k * h), g(-1 + ((k + p) * h))), (-1 + ((k + 1) * h), g(-1 + ((k + p) * h))), (-1 + ((k + 1) * h), f(-1 + ((k + p) * h)))], k, 0, m - 1) → SlicesBase
  11. Sequence(Polygon[(-1 + (k * h), f(-1 + ((k + p) * h)), 0), (-1 + (k * h), g(-1 + ((k + p) * h)), 0), (-1 + (k * h), g(-1 + ((k + p) * h)), ht(-1 + ((k + p) * h)))], k, 0, m - 1) → SidesLeft
  12. Sequence(Polygon[(-1 + ((k + 1) * h), f(-1 + ((k + p) * h)), 0), (-1 + ((k + 1) * h), g(-1 + ((k + p) * h)), 0), (-1 + ((k + 1) * h), g(-1 + ((k + p) * h)), ht(-1 + ((k + p) * h)))], k, 0, m - 1) → SidesRight
  13. Sequence(Surface[-1 + ((k + u) * h), (v * g(-1 + ((k + p) * h))) + ((1 - v) * f(-1 + ((k + p) * h))), (v * ht(-1 + ((k + p) * h))), u, 0, 1, v, 0, 1], k, 0, m - 1) → SlicesTop
  14. Sequence(Surface[-1 + ((k + u) * h), g(-1 + ((k + p) * h)), (v * ht(-1 + ((k + p) * h))), u, 0, 1, v, 0, 1], k, 0, m - 1) → SlicesBack
  15. Sequence(Area[Element[SidesLeft, k]], k, 1, m) → Areas