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Cross Sectional Solids Visualization

Cross Sectional Solids Visualization
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2015-05-27 01:41
Solid GeometryCalculusIntegralGeometric ConstructionSlider Animation
In collections Calculus Concepts
This example visualizes solids generated by cross sections protruding from the Cartesian plane. It involves core math concepts like integral calculus, volume calculation, and geometric shape construction. Key GeoGebra commands used include If, Midpoint, Polygon, CircumcircleArc, Segment, ImplicitCurve, Integral, along with sliders and input boxes to adjust cross-sectional shapes (e.g., squares, equilateral triangles, semicircles) and observe the resulting solids. Suitable for high school and above students to understand volume integrals in solid geometry and calculus.

GeoGebra Commands & Parameters

  1. If(PerpXAxis, y_2Point, x_2Point) → Relation_2Point
  2. If(PerpXAxis, y_1Point, x_1Point) → Relation_1Point
  3. Midpoint(Relation_1Point, Relation_2Point) → RelationMidpoint
  4. If(PerpXAxis, abs(y(Relation_2Point) - y(Relation_1Point)), abs(x(Relation_2Point) - x(Relation_1Point))) → ShapeBaseWidth
  5. Textfield(y_1) → y_1InputBox
  6. Textfield(y_2) → y_2InputBox
  7. Polygon(Relation_1Point, Relation_2Point, EquilateralTriangleTopPoint) → TheEquilateralTriangle, equilateraltriangleside1, equilateraltriangleside2, equilateraltriangleside3
  8. Polygon(Relation_1Point, Relation_2Point, SquareRelation_2TopPoint, SquareRelation_1TopPoint) → TheSquare, SquareBottomSide, SquareUpperSide, SquareTopSide, SquareLowerSide
  9. CircumcircleArc(Relation_1Point, SemiCircleTopPoint, Relation_2Point) → TheSemiCircleArc
  10. Segment(Relation_1Point, Relation_2Point) → TheSemiCircleBase
  11. If(PerpXAxis, RectangleHeightX, RectangleHeightY) → RectangleHeight
  12. Polygon(Relation_1Point, Relation_2Point, RectangleRelation_2TopPoint, RectangleRelation_1TopPoint) → TheRectangle, rectangleside3, rectangleside4, rectangleside2, rectangleside1
  13. If(PerpXAxis, IsoscelesTriangleHeightX, IsoscelesTriangleHeightY) → IsoscelesTriangleHeight
  14. Polygon(Relation_1Point, Relation_2Point, IsoscelesTriangleTopPoint) → TheIsoscelesTriangle, isoscelestriangleside1, isoscelestriangleside2, isoscelestriangleside3
  15. If(PerpXAxis, abs(y_2(x) - y_1(x)), abs(g(x) - f(x))) → ShapeBaseFormula
  16. If(Squares, ShapeBaseFormula(x)^(2), If[EquilateralTriangles, (sqrt(3) / 4 * ShapeBaseFormula(x)^(2)), If[SemiCircles, ((0.5 * π) * (ShapeBaseFormula(x) / 2)^(2)), If[Rectangles, (RectangleHeight(x) * ShapeBaseFormula(x)), If[IsoscelesTriangles, ((1 / 2 * ShapeBaseFormula(x)) * IsoscelesTriangleHeight(x))]]]]) → ShapeAreaFormula
  17. Textfield(x_1InputText) → x_1InputBox
  18. Textfield(x_2InputText) → x_2InputBox
  19. Textfield(RectangleHeightYInputText) → RectangleHeightInputBoxPerpYAxis
  20. Textfield(TriangleHeightYInputText) → IsoscelesTriangleHeightInputBoxPerpYAxis
  21. ImplicitCurve((-x) + f(y)) → x_1
  22. ImplicitCurve((-x) + g(y)) → x_2
  23. If(PerpXAxis, LowerXBound, LowerYBound) → LowerBound
  24. Integral(ShapeAreaFormula, LowerBound, ShapeSlider) → ShapeVolume
  25. Textfield(UpperXBound) → UpperXBoundInputBox
  26. Textfield(LowerXBound) → LowerXBoundInputBox
  27. If(PerpXAxis, UpperXBound, UpperYBound) → UpperBound
  28. Textfield(LowerYBound) → LowerYBoundInputBox
  29. Textfield(UpperYBound) → UpperYBoundInputBox
  30. Textfield(RectangleHeightX) → inputBox1
  31. Textfield(IsoscelesTriangleHeightX) → inputBox2

AI Drawing Prompt Reference

Draw a 3D solid generated by cross-sectional shapes (equilateral triangles, squares, semicircles, rectangles, isosceles triangles) protruding from the Cartesian plane. The cross-sectional shapes are adjustable via input boxes. The endpoints of the cross-sections are defined by lines, and the midpoint is used to determine the center. The area of each cross-section is calculated using a formula that depends on the shape, and the volume is computed via integration. The solid includes implicit curves defining the x-coordinates and input boxes for setting the cross-section parameters.
The prompt above was compiled from the original drawing commands and is for reference only.