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Parabola Rotated Around x-Axis

Parabola Rotated Around x-Axis
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2014-02-17 23:57
FunctionCoordinate GeometrySolid Geometry
In collections BC Calculus
This example demonstrates the solid formed by rotating the parabola [math]f(x)=4x-x^2[/math] around the x-axis. Key mathematical concepts include coordinate geometry and solid geometry. Using commands like Sequence, Vector, and CurveCartesian, the rotated solid can be constructed. Suitable for high school mathematics education and students learning solid geometry.

GeoGebra Commands & Parameters

  1. Sequence(Segment[((-4) * X) + (a * Y), (4 * X) + (a * Y)], a, -4, 4) → list1
  2. Vector(O, (5 * X)) → xx
  3. Vector(O, (5 * Y)) → yy
  4. Vector(O, (5 * Z)) → zz
  5. Sequence(Segment[((-4) * Y) + (a * X), (4 * Y) + (a * X)], a, -4, 4) → list2
  6. CurveCartesian(x(O + (t * X) + (f(t) * Y) + (0 * Z)), y(O + (t * X) + (f(t) * Y) + (0 * Z)), t, 0, 4) → a
  7. Sequence((a * Z), a, 1, 4) → list3
  8. Sequence((a * Y), a, 1, 4) → list4
  9. Sequence((a * X), a, 1, 4) → list5
  10. Sequence(CurveCartesian[x((n * X) + ((f(n) * cos(m)) * Y) + ((f(n) * sin(m)) * Z)), y((n * X) + ((f(n) * cos(m)) * Y) + ((f(n) * sin(m)) * Z)), m, 0, 6.28319], n, 0, 4, 0.1) → list9
  11. Sequence(CurveCartesian[x((n * X) + ((f(n) * cos(m)) * Y) + ((f(n) * sin(m)) * Z)), y((n * X) + ((f(n) * cos(m)) * Y) + ((f(n) * sin(m)) * Z)), n, 0, 4], m, 0, 6.283185307179586, 0.15707963267948966) → list10

AI Drawing Prompt Reference

Draw a solid formed by rotating the parabola f(x)=4x-x² around the x-axis, with x ranging from 0 to 4. First, draw the three coordinate axes (x, y, z) and add several grid lines parallel to each axis. Then plot the parabola in the xy-plane. Next, generate a series of circular cross-sections along the x-axis, each centered on the x-axis with radius equal to the function value at that x. Finally, connect these circles with smooth curves to show the surface of the rotated solid, resembling a flared horn. Keep the axes and grid lines visible for reference.
The prompt above was compiled from the original drawing commands and is for reference only.