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Volumes: Disk Method

Volumes: Disk Method
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2015-12-22 15:17
Solid GeometryCalculusIntegral
This example demonstrates how to calculate the volume of a solid of revolution using the disk method. Key mathematical concepts include integration and solid geometry. Using GeoGebra commands such as CurveCartesian, Surface, and Intersect, the solid can be constructed. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Textfield(f) → Function
  2. Textfield(a) → Textfeld2
  3. Textfield(b) → Textfeld3
  4. CurveCartesian(t, (f(t) * cos(φ)), (f(t) * sin(φ)), t, a, b) → k_x
  5. Ray((0, c), Vector[(1, 0)]) → g_1
  6. Intersect(f, g_1, (1, f(1))) → I
  7. Ray((0, d), Vector[(1, 0)]) → g_2
  8. Intersect(f, g_2, (d, f(d))) → J
  9. CurveCartesian((t * cos(ϕ)), f(t), (t * sin(ϕ)), t, x(I), x(J)) → k_y
  10. CurveCartesian(t, f(t), 0, t, -5, 5) → k
  11. Textfield(c) → Textfeld4
  12. Textfield(d) → Textfeld5
  13. Segment(A, G) → s_1
  14. Segment(B, F) → s_2
  15. Surface(u, (f(u) * cos(w)), (f(u) * sin(w)), u, a, b, w, 0, φ) → f_x
  16. Surface((uu * cos(ww)), f(uu), (uu * sin(ww)), uu, x(I), x(J), ww, 0, ϕ) → f_y
  17. Invert(f) → f_u
  18. Segment(C, I) → s_3
  19. Segment(D, J) → s_4
  20. CurveCartesian(x(B), t, t, y(B), f(x(B))) → LineCapRight
  21. CurveCartesian(x(A), t, t, y(A), f(x(A))) → LineCapLeft
  22. Surface(x(A), y(A) + ((s * (f(x(A)) - y(A))) * cos(t)), ((s * (f(x(A)) - y(A))) * sin(t)), s, 0, 1, t, 0, φ) → CapLeft
  23. Surface(x(B), y(B) + ((s * (f(x(B)) - y(B))) * cos(t)), ((s * (f(x(B)) - y(B))) * sin(t)), s, 0, 1, t, 0, φ) → CapRight
  24. CurveCartesian(t, y(C), t, x(C), f_u(y(C))) → LineCapBottom
  25. CurveCartesian(t, y(D), t, x(D), f_u(y(D))) → LineCapTop

AI Drawing Prompt Reference

Draw a demonstration of calculating the volume of a solid of revolution using the disk method. Include a function curve and two perpendicular rays to the x-axis, with their intersections marked as I and J. Based on the curve and rays, sketch the surface and cross-sections of the solid, adding helper line segments and surface caps to clearly illustrate the geometric meaning of the disk method.
The prompt above was compiled from the original drawing commands and is for reference only.