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3D Volume by Rotating a Function

3D Volume by Rotating a Function
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2013-06-08 19:41
FunctionSolid GeometryCalculusDerivativeIntegral
In collections BC Calculus
This example demonstrates the process of generating 3D volume by rotating a function. Key mathematical concepts include rotation, volume, and quadratic functions. It utilizes commands such as Vector, Text, CurveCartesian, Point, Sequence, Function, Segment, Textfield, and Integral in GeoGebra. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Vector(O, X) → u
  2. Vector(O, Y) → v
  3. Vector(O, Z) → w
  4. Vector(O - L_{Axis} * u, O + L_{Axis} * u) → u_L
  5. Vector(O - L_{Axis} * v, O + L_{Axis} * v) → v_L
  6. Vector(O - L_{Axis} * w, O + L_{Axis} * w) → w_L
  7. Text("x", O + L_{Axis} * u) → text1
  8. Text("y", O + L_{Axis} * v) → text2
  9. Text("z", O + L_{Axis} * w) → text3
  10. CurveCartesian(x(O + t * u + f(t) * v), y(O + t * u + f(t) * v), t, -L_{Axis}, L_{Axis}) → a
  11. Point(xAxis) → A
  12. Point(xAxis) → B
  13. CurveCartesian(x(O + t * u + f(t) * cos(Rotation) * v + f(t) * sin(Rotation) * w), y(O + t * u + f(t) * cos(Rotation) * v + f(t) * sin(Rotation) * w), t, -L_{Axis}, L_{Axis}) → b
  14. CurveCartesian(x(O + x(A) * u + f(x(A)) * cos(s) * v + f(x(A)) * sin(s) * w), y(O + x(A) * u + f(x(A)) * cos(s) * v + f(x(A)) * sin(s) * w), s, 0, Rotation) → d
  15. CurveCartesian(x(O + x(B) * u + f(x(B)) * cos(s) * v + f(x(B)) * sin(s) * w), y(O + x(B) * u + f(x(B)) * cos(s) * v + f(x(B)) * sin(s) * w), s, 0, Rotation) → e
  16. Sequence(CurveCartesian[x(O + t * u + f(t) * cos(s) * v + f(t) * sin(s) * w), y(O + t * u + f(t) * cos(s) * v + f(t) * sin(s) * w), t, x(A), x(B)], s, 0, Rotation, 15*°) → liste3
  17. Function(f, x(A), x(B)) → g
  18. CurveCartesian(x(O + t * u + f(t) * v), y(O + t * u + f(t) * v), t, x(A), x(B)) → h
  19. CurveCartesian(x(O + t * u + f(t) * cos(Rotation) * v + f(t) * sin(Rotation) * w), y(O + t * u + f(t) * cos(Rotation) * v + f(t) * sin(Rotation) * w), t, x(A), x(B)) → i_1
  20. Sequence(CurveCartesian[x(O + t * u + f(t) * cos(s) * v + f(t) * sin(s) * w), y(O + t * u + f(t) * cos(s) * v + f(t) * sin(s) * w), s, 0, Rotation], t, x(A), x(B), (x(B) - x(A)) / Parts) → liste4
  21. Vector(O - L_{Axis} * u, O + L_{Axis} * u) → z
  22. Segment(O + x(A) * u + k / Parts * (x(B) - x(A)) * u, O + x(A) * u + k / Parts * (x(B) - x(A)) * u + f(x(A) + k / Parts * (x(B) - x(A))) * v) → l
  23. Textfield(f) → ChampTexte1
  24. Integral(g, x(A), x(B)) → j

AI Drawing Prompt Reference

Draw a 3D volume generated by rotating the function f(x) around the x-axis. First, set up the coordinate system and define the x, y, and z axes. Then, draw the graph of the function f(x) and rotate it around the x-axis to form the 3D volume. Finally, calculate and display the integral value of the volume.
The prompt above was compiled from the original drawing commands and is for reference only.