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Washer Method for Volume of Revolution

Washer Method for Volume of Revolution
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2016-05-10 21:45
Solid GeometryCalculusIntegral
This example demonstrates the washer method for calculating the volume of a solid of revolution. Key mathematical concepts include integration in calculus. By rotating a function graph, constructing washers, and calculating their volumes, this example uses commands like Invert, Line, Rotate, Textfield, Segment, and Surface in GeoGebra. Suitable for high school math education and learning.

GeoGebra Commands & Parameters

  1. Invert(g1) → i1
  2. Invert(g2) → i2
  3. Line(A, B) → L
  4. Rotate(f1, θ, L) → p1
  5. Line(C, D) → M
  6. Textfield(c) → inputBox1
  7. Textfield(d) → inputBox2
  8. Textfield(g1) → inputBox3
  9. Textfield(b) → inputBox5
  10. Textfield(a) → inputBox6
  11. Rotate(f2, θ, L) → p2
  12. Textfield(g2) → inputBox4
  13. Segment(E, F) → EF
  14. Segment(G, H) → GH
  15. Rotate(E, θ, L) → E'
  16. Rotate(F, θ, L) → F'
  17. Segment(E', F') → EF'
  18. Rotate(G, θ, L) → G'
  19. Rotate(H, θ, L) → H'
  20. Segment(G', H') → GH'
  21. Surface(x(F), ((y(F) - c) * cos(t)) + c + ((s * (f1(x(E)) - f2(x(F)))) * cos(t)), ((y(F) - c) * sin(t)) + ((s * (f1(x(E)) - f2(x(F)))) * sin(t)), s, 0, 1, t, 0, θ) → lcap
  22. Surface(x(H), ((y(H) - c) * cos(t)) + c + ((s * (f1(x(G)) - f2(x(H)))) * cos(t)), ((y(H) - c) * sin(t)) + ((s * (f1(x(G)) - f2(x(H)))) * sin(t)), s, 0, 1, t, 0, θ) → rcap
  23. Surface(u, ((f1(u) - c) * cos(w)) + c, ((f1(u) - c) * sin(w)), u, a, b, w, 0, θ) → SurfaceOuterAboutHoriz
  24. Surface(u, ((f2(u) - c) * cos(w)) + c, ((f2(u) - c) * sin(w)), u, a, b, w, 0, θ) → SurfaceInnerAboutHoriz
  25. Surface(((i2(u) - d) * cos(w)) + d, u, ((i2(u) - d) * sin(w)), u, y(F), y(H), w, 0, θ) → SurfaceInnerAboutVert
  26. Surface(((i1(u) - d) * cos(w)) + d, u, ((i1(u) - d) * sin(w)), u, y(E), y(G), w, 0, θ) → SurfaceOuterAboutVert
  27. Rotate(E, (-θ), M) → E'_1
  28. Rotate(F, (-θ), M) → F'_1
  29. Rotate(G, (-θ), M) → G'_1
  30. Rotate(H, (-θ), M) → H'_1
  31. Segment(E'_1, F'_1) → E'1F'1
  32. Segment(G'_1, H'_1) → G'1H'1
  33. Rotate(f1, (-θ), M) → q1
  34. Rotate(f2, (-θ), M) → q2
  35. Surface(((x(G) - d) * cos(w)) + d, u, ((x(G) - d) * sin(w)), u, y(G), y(H), w, 0, θ) → ocap
  36. Surface(((x(E) - d) * cos(w)) + d, u, ((x(E) - d) * sin(w)), u, y(E), y(F), w, 0, θ) → icap
  37. Surface((Slice * x(H)), ((y(H) - c) * cos(t)) + c + ((s * (Slice * (f1(x(G)) - f2(x(H))))) * cos(t)), ((y(H) - c) * sin(t)) + ((s * (Slice * (f1(x(G)) - f2(x(H))))) * sin(t)), s, 0, 1, t, 0, θ) → lslice

AI Drawing Prompt Reference

Draw a solid of revolution using the washer method to calculate its volume. First, plot two function graphs f1 and f2, then rotate them around the x-axis to form outer and inner washers. Draw vertical line segments, rotate them to form the vertical cross-sections of the inner and outer washers, and finally use the surface command to create the solid.
The prompt above was compiled from the original drawing commands and is for reference only.