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Volume by Shells

Volume by Shells
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2016-03-18 03:18
TrigonometryCalculusIntegral
In collections AP Calculus volume problems
This example demonstrates a typical problem from the AP Calculus test, illustrating the calculation of the volume of a solid of revolution using integration. Key mathematical concepts include integration, volume of solids of revolution, and trigonometric functions. Using GeoGebra commands such as Sequence, Rotate, and Integral, the formation process of the solid of revolution is dynamically shown. Suitable for high school calculus education and learning.

GeoGebra Commands & Parameters

  1. Sequence(Polygon[((k * hx), g(((k + p) * hx))), (((k + 1) * hx), g(((k + p) * hx))), (((k + 1) * hx), f(((k + p) * hx))), ((k * hx), f(((k + p) * hx)))], k, 0, m - 1) → VRectangles
  2. Rotate(B, α, a) → B'
  3. Integral((((2 * π) * (1 + x)) * (f(x) - g(x))), 0, 9) → VolShellIntegral
  4. Integral((π * ((1 + gI(y))^(2) - (1 + fI(y))^(2))), 0, 3) → VolWasherIntegral
  5. Sequence(Polygon[(gI(((k + p) * hy)), (k * hy)), (fI(((k + p) * hy)), (k * hy)), (fI(((k + p) * hy)), ((k + 1) * hy)), (gI(((k + p) * hy)), ((k + 1) * hy))], k, 0, m - 1) → HRectangles
  6. Rotate(HRectangles, (-α), a) → HRectangles'
  7. Rotate(VRectangles, (-α), a) → VRectangles'
  8. If(togSurf ≟ 1, "Hide surface", "Show surface") → ShowSurface
  9. Surface(-1 + ((1 + gI(u)) * cos(β)), u, ((1 + gI(u)) * sin(β)), u, 0, 3, β, 0, (2 * π)) → Outer
  10. Surface(-1 + ((1 + fI(u)) * cos(β)), u, ((1 + fI(u)) * sin(β)), u, 0, 3, β, 0, (2 * π)) → Inner
  11. If(togSH ≟ 1, "Use washers", "Use shells") → ShellWasher
  12. Surface(-1 + ((1 + u) * cos(β)), f(((m - 1 + p) * hx)), ((1 + u) * sin(β)), u, ((m - 1) * hx), (m * hx), β, 0, α) → ShellsBack1
  13. Surface(-1 + ((1 + u) * cos(β)), (g(m - 1 + p) * hx), ((1 + u) * sin(β)), u, ((m - 1) * hx), (m * hx), β, 0, α) → ShellsFront1
  14. Surface(-1 + ((1 + ((m - 1) * hx)) * cos(β)), v, ((1 + ((m - 1) * hx)) * sin(β)), v, g(((m - 1 + p) * hx)), f(((m - 1 + p) * hx)), β, 0, α) → ShellsInner1
  15. Surface(-1 + ((1 + (m * hx)) * cos(β)), v, ((1 + (m * hx)) * sin(β)), v, g(((m - 1 + p) * hx)), f(((m - 1 + p) * hx)), β, 0, α) → ShellsOuter1
  16. Polygon((gI(((m - 1 + p) * hy)), ((m - 1) * hy)), (fI(((m - 1 + p) * hy)), ((m - 1) * hy)), (fI(((m - 1 + p) * hy)), (m * hy)), (gI(((m - 1 + p) * hy)), (m * hy))) → RectWash, b, c, d, e
  17. Polygon((((m - 1) * hx), g(((m - 1 + p) * hx))), (((m - 1) * hx), f(((m - 1 + p) * hx))), ((m * hx), f(((m - 1 + p) * hx))), ((m * hx), g(((m - 1 + p) * hx)))) → RectShell, h, i, j, k
  18. Surface(-1 + ((1 + fI(((m - 1 + p) * hy))) * cos(β)), v, ((1 + fI(((m - 1 + p) * hy))) * sin(β)), v, ((m - 1) * hy), (m * hy), β, 0, α) → WasherInner1
  19. Surface(-1 + ((1 + gI(((m - 1 + p) * hy))) * cos(β)), v, ((1 + gI(((m - 1 + p) * hy))) * sin(β)), v, ((m - 1) * hy), (m * hy), β, 0, α) → WashOuter1
  20. Surface(-1 + ((1 + u) * cos(β)), ((m - 1) * hy), ((1 + u) * sin(β)), u, fI(((m - 1 + p) * hy)), gI(((m - 1 + p) * hy)), β, 0, α) → WashFront1
  21. Surface(-1 + ((1 + u) * cos(β)), (m * hy), ((1 + u) * sin(β)), u, fI(((m - 1 + p) * hy)), gI(((m - 1 + p) * hy)), β, 0, α) → WashBack1
  22. RectangleSum((((2 * π) * (1 + x)) * (sqrt(x) - x / 3)), 0, (m * hx), m, p) → VolShells
  23. RectangleSum((π * ((1 + (3 * y))^(2) - (1 + y^(2))^(2))), 0, (m * hy), m, p) → VolWashers
  24. Sequence(Surface[-1 + ((1 + u) * cos(β)), ((k + 1) * hy), ((1 + u) * sin(β)), u, fI(((k + p) * hy)), gI(((k + p) * hy)), β, 0, α], k, 0, m - 1) → WashersBack
  25. Sequence(Surface[-1 + ((1 + fI(((k + p) * hy))) * cos(β)), ((k + v) * hy), ((1 + fI(((k + p) * hy))) * sin(β)), v, 0, 1, β, 0, α], k, 0, m - 1) → WashInners
  26. Sequence(Surface[-1 + ((1 + (u * fI(((k + p) * hy))) + ((1 - u) * gI(((k + p) * hy)))) * cos(β)), ((k + u) * hy), ((1 + (u * fI(((k + p) * hy))) + ((1 - u) * gI(((k + p) * hy)))) * sin(β)), u, 0, 1, β, 0, α], k, 0, m - 1) → WasherFronts
  27. Sequence(Surface[-1 + ((1 + gI(((k + p) * hy))) * cos(β)), ((k + v) * hy), ((1 + gI(((k + p) * hy))) * sin(β)), v, 0, 1, β, 0, α], k, 0, m - 1) → WasherOuters
  28. Sequence(Surface[-1 + ((1 + u) * cos(β)), f(((k + p) * hx)), ((1 + u) * sin(β)), u, (k * hx), ((k + 1) * hx), β, 0, α], k, 0, m - 1) → ShellsBack
  29. Sequence(Surface[-1 + ((1 + u) * cos(β)), f(((k + p) * hx)), ((1 + u) * sin(β)), u, (k * hx), ((k + 1) * hx), β, 0, α], k, 0, m - 1) → ShellsFront
  30. Sequence(Surface[-1 + ((1 + (k * hx)) * cos(β)), (v * g(((k + p) * hx))) + ((1 - v) * f(((k + p) * hx))), ((1 + (k * hx)) * sin(β)), v, 0, 1, β, 0, α], k, 0, m - 1) → ShellsSidesLeft
  31. Sequence(Surface[-1 + ((1 + ((k + 1) * hx)) * cos(β)), (v * g(((k + p) * hx))) + ((1 - v) * f(((k + p) * hx))), ((1 + ((k + 1) * hx)) * sin(β)), v, 0, 1, β, 0, α], k, 0, m - 1) → ShellsSidesRight

AI Drawing Prompt Reference

Create a visualization demonstrating the calculation of the volume of a solid of revolution using integration, suitable for AP Calculus education. Start by drawing several rectangles, then rotate them to dynamically form the shape of the solid. Next, calculate and visualize the volume using both the disk method and the shell method, showing the integral regions and results for each. Finally, display the surfaces and cross-sections of the solid based on the chosen calculation method.
The prompt above was compiled from the original drawing commands and is for reference only.