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圆盘法计算旋转体体积

圆盘法计算旋转体体积
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2016-03-20 05:49
函数立体几何微积分积分
所属专题 AP Calculus volume problems
这个范例演示了如何使用圆盘法计算旋转体的体积。涉及的核心数学概念是定积分和旋转体体积计算。作图要点包括使用Sequence、Rotate、Integral、If、Surface、Polygon、RectangleSum等GeoGebra命令和对象来构建图形和计算体积。适合高中和大学低年级学生学习微积分和立体几何。

GeoGebra 命令及参数

  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