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不同截面立体的体积

不同截面立体的体积
几何画板及绘图内容加载中，请稍等……
2019-04-12 07:51
三角函数解析几何立体几何微积分几何作图
①本范例演示如何通过不同截面绘制立体图形并计算其体积；②涉及核心数学概念包括体积计算、直角三角形、三角函数、相似三角形、立体几何；③构造过程中运用了Segment、Point、Polygon、IntegralBetween等GeoGebra命令；④适合高中及以上学生及数学爱好者用于理解立体几何体积计算方法。

GeoGebra 命令及参数

  1. SelectedElement(C) → text1
  2. IndexOf(text1, C) → a
  3. Intersect(f, g, 1) → A
  4. Intersect(f, g, 2) → B
  5. Segment((x(A), -3), (x(B), -3)) → h
  6. Point(h) → D
  7. Segment(E, F) → i
  8. CurveCartesian(t, g(t), p(t), t, x(A), x(B)) → b
  9. CurveCartesian(t, f(t), p(t), t, x(A), x(B)) → c
  10. Polygon(E, F, (x(F), f(x(F)), p(x(F))), (x(F), g(x(F)), p(x(F)))) → q1, e, f_1, j, k
  11. Surface(t, g(t), (u * p(t)), t, x(A), x(B), u, 0, 1) → d
  12. Surface(t, f(t), (u * p(t)), t, x(A), x(B), u, 0, 1) → l
  13. Surface(t, (u * g(t)) + ((1 - u) * f(t)), p(t), t, x(A), x(B), u, 0, 1) → m
  14. Polygon(E, F, F + (0, 0, e), E + (0, 0, e)) → q2, e_1, f_2, n, q
  15. CurveCartesian(t, g(t), f(t) - g(t), t, x(A), x(B)) → o
  16. CurveCartesian(t, f(t), f(t) - g(t), t, x(A), x(B)) → r
  17. Surface(t, g(t), (u * (f(t) - g(t))), t, x(A), x(B), u, 0, 1) → s
  18. Surface(t, f(t), (u * (f(t) - g(t))), t, x(A), x(B), u, 0, 1) → t
  19. Surface(t, (u * f(t)) + ((1 - u) * g(t)), f(t) - g(t), t, x(A), x(B), u, 0, 1) → u
  20. Midpoint(i) → G
  21. Surface(t, ((f(t) - g(t)) / 2 * cos(u)) + (f(t) + g(t)) / 2, ((f(t) - g(t)) / 2 * sin(u)), t, x(A), x(B), u, 0, (2 * π)) → v
  22. OrthogonalPlane(G, i) → w
  23. PointIn(w) → H
  24. Line(G, H) → g_1
  25. Circle(G, (f(x(G)) - g(x(G))) / 2, g_1) → c_1
  26. IntegralBetween(f, g, x(A), x(B)) → a_1
  27. CurveCartesian(t, (f(t) + g(t)) / 2, ((f(t) - g(t)) / 2 * sqrt(3)), t, x(A), x(B)) → bb
  28. Surface(t, (u * g(t)) + ((1 - u) * y(bb(t))), ((1 - u) * z(bb(t))), t, x(A), x(B), u, 0, 1) → b_1
  29. Surface(t, (u * f(t)) + ((1 - u) * y(bb(t))), ((1 - u) * z(bb(t))), t, x(A), x(B), u, 0, 1) → d_1
  30. Rotate(Polygon[E, F, 3], π / 2, e) → t1
  31. CurveCartesian(t, f(t), f(t) - g(t), t, x(A), x(B)) → hh
  32. Polygon(E, F, F + (0, 0, f(x(F)) - g(x(F)))) → t2, i_1, e_2, f_3
  33. Surface(t, (u * g(t)) + ((1 - u) * f(t)), ((1 - u) * z(hh(t))), t, x(A), x(B), u, 0, 1) → h_1
  34. Surface(t, f(t), (u * z(hh(t))), t, x(A), x(B), u, 0, 1) → j_1
  35. Surface(t, ((f(t) - g(t)) / 2 * cos(u)) + (f(t) + g(t)) / 2, ((f(t) - g(t)) / 2 * sin(u)), t, x(A), x(B), u, 0, π) → k_1
  36. Rotate(Semicircle[E, F], π / 2, e) → p_1

AI 绘图提示语参考

绘制一个立体图形，该图形由不同横截面组成。首先绘制两条曲线 f 和 g，然后在它们之间创建一系列平行于 x 轴的矩形横截面，接着使用这些横截面构建立体表面。最后，计算并显示这两个曲线围成区域的体积。
以上提示语为根据原始绘图命令整理而来，仅供参考。