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可调节的半圆柱体：平行于y轴的半圆横截面

可调节的半圆柱体：平行于y轴的半圆横截面
几何画板及绘图内容加载中，请稍等……
2019-04-02 22:43
函数立体几何微积分
这个范例展示了如何通过调整上下函数来构造一个具有半圆横截面的可调节立体图形。核心数学概念包括微积分中的积分和立体几何中的体积计算。通过使用GeoGebra命令如CurveCartesian、Surface和Rotate等，可以直观地理解横截面如何影响整个立体的形状和体积。适合高中及以上学生学习立体几何和微积分。

GeoGebra 命令及参数

  1. Segment(H, I) → s_1
  2. Midpoint(s_1) → E
  3. Segment(J, K) → q_1
  4. Midpoint(q_1) → F
  5. Intersect(upper, h, (0, 5)) → B
  6. Intersect(lower, h, (0, -8)) → C
  7. Segment(C, B) → q
  8. Intersect(upper, q, (0, 5)) → G
  9. Midpoint(C, G) → T
  10. Line(B, D) → i
  11. Sphere(B, d) → e
  12. Segment(C, B) → p
  13. Textfield(upper) → InputBox1
  14. Textfield(lower) → InputBox2
  15. Textfield(a) → InputBox3
  16. Textfield(b) → InputBox4
  17. Segment(C, G) → k_1
  18. Midpoint(J, K) → Z
  19. Midpoint(Q, P) → A_1
  20. CurveCartesian(b, y(Z) + ((y(J) - y(Z)) * cos(u)), ((y(J) - y(Z)) * sin(u)), u, 0, pi) → f
  21. CurveCartesian(a, y(A_1) + ((y(Q) - y(A_1)) * cos(u)), ((y(Q) - y(A_1)) * sin(u)), u, 0, pi) → c
  22. Surface(u, (0.5 * (upper(u) + lower(u))) + ((0.5 * (upper(u) - lower(u))) * cos(v)), ((0.5 * (upper(u) - lower(u))) * sin(v)), u, a, b, v, 0, pi) → g
  23. Midpoint(C, G) → W
  24. CurveCartesian(x(W), (0.5 * (y(G) + y(C))) + ((0.5 * (y(G) - y(C))) * cos(u)), ((0.5 * (y(G) - y(C))) * sin(u)), u, 0, pi) → j
  25. Line(W, B_1) → k
  26. Sphere(W, G) → l
  27. Intersect(k, l) → C_1, D_1
  28. Rotate(D_1, ((-pi)) / 2, W) → E_1
  29. Rotate(D_1, pi / 2, W) → F_1
  30. CircleArc(W, E_1, F_1, xOyPlane) → r
  31. Line(W, E_1) → m
  32. Rotate(r, pi / 2, W, m) → r'
  33. Midpoint(q_1) → G_1
  34. Midpoint(s_1) → H_1
  35. Line(G_1, G_1 + (1, 0, 0)) → n
  36. Line(H_1, H_1 + (1, 0, 0)) → s
  37. Sphere(G_1, J) → o
  38. Intersect(n, o) → I_1, J_1
  39. Sphere(H_1, Q) → t
  40. Intersect(s, t) → K_1, L_1
  41. Rotate(J_1, ((-pi)) / 2, G_1) → M_1
  42. Rotate(J_1, pi / 2, G_1) → N_1
  43. CircleArc(G_1, M_1, N_1, xOyPlane) → c_1
  44. Rotate(c_1, ((-pi)) / 2, G_1, Line[G_1, N_1]) → e_1
  45. Rotate(L_1, ((-pi)) / 2, H_1) → O_1
  46. Rotate(L_1, pi / 2, H_1) → P_1
  47. CircleArc(H_1, O_1, P_1, xOyPlane) → d_1
  48. Rotate(d_1, ((-pi)) / 2, H_1, Line[H_1, Q]) → f_1