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旋转体体积的黎曼和可视化：中值逼近

旋转体体积的黎曼和可视化：中值逼近
几何画板及绘图内容加载中，请稍等……
2021-01-30 11:36
数列立体几何微积分积分几何作图
该范例演示如何用黎曼和思想计算由平面曲线绕坐标轴旋转形成的旋转体体积，并展示分区数量变化时近似值的收敛趋势。核心数学概念包括旋转体体积公式的思想、黎曼和/中值法（用每个区间的中点函数值代表高度）、以及通过圆盘法将“截面面积”累加为体积。作图要点是利用构造摘要中的 If 与 Surface 来组织分段条件与生成旋转曲面，借助 Sequence 来创建分区序列与用于控制细分数 n 的参数，再将生成的截面/曲面对象（如 a、nn、l1–l14、l2–l12 等用于描述长度或分割区间，h、r、f、g 等用于定义半径或函数值，a_1 用于汇总结果）用于可视化体积逼近。适合高中到大学初阶微积分课程中“积分思想、旋转体、黎曼和”相关教学与探究活动，也适合作为GeoGebra建模练习。

GeoGebra 命令及参数

  1. If(a < 10, 1, If[a < 20, 2, If[a < 30, 4, If[a < 40, 5, If[a < 50, 8, 4]]]]) → nn
  2. If(a < 10, a, If[a < 20, a - 10, If[a < 30, a - 20, If[a < 40, a - 30, If[a < 50, a - 40, If[a < 60, a - 50, If[a < 70, a - 60, If[a < 80, a - 70, If[a < 90, a - 80, a - 90]]]]]]]]) → b
  3. Surface((r * cos(t)), (r * sin(t)), r^(2), r, 0, 2, t, 0, (b * 0.63)) → c
  4. Surface((r * cos(t)), (r * sin(t)), 4, r, 0, 2, t, 0, (b * 0.63)) → d
  5. Surface((r * cos(t)), (r * sin(t)), 1, r, 0, 1, t, 0, (b * 0.63)) → e
  6. Sequence(Surface[(u * cos(t)), (u * sin(t)), ((m - 1) * 4 / nn), u, 0, sqrt(((m - 1) * 4 / nn)), t, 0, (b * 0.63)], m, 1, nn) → l2
  7. Sequence(Surface[(sqrt((4 * (m - 1) / nn)) * cos(t)), (sin(t) * sqrt((4 * (m - 1) / nn))), z, z, (4 * (m - 1) / nn), (4 * m / nn), t, 0, (b * 0.63)], m, 1, nn) → l1
  8. Sequence(Surface[u, 0, z, u, 0, sqrt((4 * (m - 1) / nn)), z, ((m - 1) * 4 / nn), (m * 4 / nn)], m, 1, nn) → l3
  9. Sequence(Surface[(u * cos((b * 0.63))), (u * sin((b * 0.63))), z, u, 0, sqrt((4 * (m - 1) / nn)), z, ((m - 1) * 4 / nn), (m * 4 / nn)], m, 1, nn) → l4
  10. Surface(u, 0, z, z, 1, 1 + dd, u, 0, 1) → h
  11. Surface((r * cos(t)), (r * sin(t)), 1 + dd, r, 0, 1, t, 0, (b * 0.63)) → f
  12. Surface(cos(t), sin(t), z, z, 1, 1 + dd, t, 0, (b * 0.63)) → g
  13. Surface((r * cos((b * 0.63))), (r * sin((b * 0.63))), z, r, 0, 1, z, 1, 1.2) → i
  14. Sequence(Surface[u, 0, z, u, (2 * (m - 1) / nn), (2 * m / nn), z, 0, ((2 * (m - 1) / nn))^(2)], m, 1, nn) → l5
  15. Sequence(Surface[u, ((((m - 1) * 2 / nn))^(2) * sin(t)), ((((m - 1) * 2 / nn))^(2) * cos(t)), u, ((m - 1) * 2 / nn), (m * 2 / nn), t, 0, (b * 0.63)], m, 1, nn) → l7
  16. Sequence(Surface[u, (r * sin((b * 0.63))), (r * cos((b * 0.63))), u, (2 * (m - 1) / nn), (2 * m / nn), r, 0, ((2 * (m - 1) / nn))^(2)], m, 1, nn) → l6
  17. Surface(u, (u^(2) * sin(t)), (u^(2) * cos(t)), u, 0, 2, t, 0, (b * 0.63)) → j
  18. Sequence(Surface[((m - 1) * 2 / nn), (r * sin(t)), (r * cos(t)), r, 0, (((m - 1) * 2 / nn))^(2), t, 0, (b * 0.63)], m, 1, nn) → l8
  19. Surface(2, (r * sin(t)), (r * cos(t)), r, 0, 4, t, 0, (b * 0.63)) → k
  20. Surface(u, ((1 - u^(2)) * sin(t - 3.14)), 1 + ((1 - u^(2)) * cos(t - 3.14)), u, 0, 1, t, 0, (0.63 * b)) → l
  21. Surface(0, (r * sin(t - 3.14)), 1 + (r * cos(t - 3.14)), r, 0, 1, t, 0, (b * 0.63)) → m
  22. Surface(u, 0, If[z > u^(2), z], u, 0, 1, z, 0, 1) → o
  23. Surface(u, If[r > u^(2), ((1 - r) * sin((b * 0.63) - 3.14))], If[r > u^(2), 1 + ((1 - r) * cos((b * 0.63) - 3.14))], u, 0, 1, r, 0, 1) → p
  24. Sequence(Surface[(m * 1 / nn), (r * sin(t - 3.14)), 1 + (r * cos(t - 3.14)), r, 0, 1 - (m / nn)^(2), t, 0, (b * 0.63)], m, 1, nn) → l9
  25. Sequence(Surface[u, ((1 - (m / nn)^(2)) * sin(t - 3.14)), 1 + ((1 - (m / nn)^(2)) * cos(t - 3.14)), u, (m - 1) / nn, m / nn, t, 0, (b * 0.63)], m, 1, nn) → l10
  26. Surface(2 + ((2 - r) * sin(t - 1.62)), ((2 - r) * cos(t - 1.62)), 0, r, 0, 2, t, 0, (b * 0.63)) → r
  27. Surface(2 + ((2 - r) * sin(t - 1.62)), ((2 - r) * cos(t - 1.62)), r^(2), r, 0, 2, t, 0, (b * 0.63)) → q
  28. Sequence(Surface[2 + (r * sin(t - 1.62)), (r * cos(t - 1.62)), ((4 * m)) / nn, r, 0, 2 - sqrt(((4 * m)) / nn), t, 0, (b * 0.63)], m, 1, nn) → l11
  29. Sequence(Surface[2 + ((2 - sqrt(((4 * m)) / nn)) * sin(t - 1.62)), ((2 - sqrt((4 * m / nn))) * cos(t - 1.62)), z, t, 0, (b * 0.62), z, ((m - 1) * 4 / nn), (m * 4 / nn)], m, 1, nn) → l12
  30. Surface((r * cos(t)), (r * sin(t)), r^(2), r, 1, 2, t, 0, (b * 0.62)) → s
  31. Surface(cos(t), sin(t), z, z, 1, 4, t, 0, (b * 0.62)) → t
  32. Surface(u, 0, If[z > u^(2), z], u, 1, 2, z, 1, 4) → u
  33. Surface((r * cos((b * 0.62))), (r * sin((b * 0.62))), If[z > r^(2), z], r, 1, 2, z, 1, 4) → v
  34. Surface((r * cos(t)), (r * sin(t)), 4, r, 1, 2, t, 0, (b * 0.62)) → w
  35. Sequence(Surface[(r * cos(t)), (r * sin(t)), 1 + (m * 3 / nn), r, 1, sqrt(1 + (m * 3 / nn)), t, 0, (b * 0.63)], m, 1, nn) → l13
  36. Sequence(Surface[(sqrt(1 + ((m - 1) * 3 / nn)) * cos(t)), (sqrt(1 + ((m - 1) * 3 / nn)) * sin(t)), z, t, 0, (b * 0.63), z, 1 + ((m - 1) * 3 / nn), 1 + (m * 3 / nn)], m, 1, nn) → l14
  37. Surface((r * cos(t)), (r * sin(t)), r^(2), r, 0, 2, t, 0, (b * 0.63)) → a_1
  38. Surface((r * cos(t)), (r * sin(t)), (2 * r), r, 0, 2, t, 0, (b * 0.63)) → b_1
  39. Sequence(Surface[u_{1}, 0, z, u_{1}, (2 * sqrt(m / nn)), (2 * sqrt((m + 1) / nn)), z, (4 * m / nn), (4 * (m + 1) / nn)], m, 0, nn - 1) → l15
  40. Sequence(Surface[(r * cos(t)), (r * sin(t)), 1 + (m * 3 / nn), r, 0, 1, t, 0, (b * 0.63)], m, 1, nn) → l20
  41. Sequence(Surface[cos(t), sin(t), z, t, 0, (b * 0.63), z, 1 + ((m - 1) * 3 / nn), 1 + (m * 3 / nn)], m, 1, nn) → l16
  42. Sequence(Surface[u_{1}, 0, z, u_{1}, 0, 1, z, 1 + ((m - 1) * 3 / nn), 1 + (m * 3 / nn)], m, 1, nn) → l17
  43. Sequence(Surface[(r * cos((b * 0.62))), (r * sin((b * 0.62))), z, r, 0, 1, z, 1 + (3 * (m - 1) / nn), 1 + (3 * m / nn)], m, 1, nn) → l18

AI 绘图提示语参考

绘制一个通过黎曼和近似体积的旋转体图形。图形由多个三维表面组成，每个表面代表分割区间上的近似体积元素。构造顺序为先创建旋转体，再通过序列生成多个表面，逐步细化近似。关键对象包括半径r、分割数n、函数f和g，以及表面c、d、e等。图形展示体积估计随分割增加而改进的过程。
以上提示语为根据原始绘图命令整理而来，仅供参考。