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截面法体积计算

截面法体积计算
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2017-04-20 04:42
函数立体几何微积分积分几何作图
本范例演示通过截面法计算体积的过程，涉及核心数学概念包括体积、比例、几何构造、切线函数、随机过程或随机过程、反射。作图要点包括使用 GeoGebra 命令如 Sequence（数列）、Stretch（拉伸）、Line（直线）、Textfield（文本框）、Polygon（多边形）、Segment（线段）、Point（点）、OrthogonalLine（垂线）；涉及对象如 b, a, n, g, value_g, f, value_f, xyRotate, xyTilt, A, axis_x, B, axis_y, inputBox1, inputBox2, inputBox3, inputBox4, f_, g_, height, inputBox3, base_{length}, base_f, base_g, top_f, top_g, side_f, side_g, base, top_{rectangle}, face_b, c, d, e, h, face_a, i, j, k, l。适合高中及以上学生学习和理解体积计算方法。

GeoGebra 命令及参数

  1. Sequence((k, g(k)), k, a, b, (b - a) / (n - 1)) → value_g
  2. Sequence((k, f(k)), k, a, b, (b - a) / (n - 1)) → value_f
  3. Stretch((cos((-xyRotate)), sin((-xyRotate))), y = 0, cos(xyTilt)) → A
  4. Line((0, 0), A) → axis_x
  5. Stretch((cos((-xyRotate) + π / 2), sin((-xyRotate) + π / 2)), y = 0, cos(xyTilt)) → B
  6. Line((0, 0), B) → axis_y
  7. Textfield(f) → inputBox1
  8. Textfield(g) → inputBox2
  9. Textfield(a) → inputbox3
  10. Textfield(b) → inputBox4
  11. Stretch(Rotate[f, (-xyRotate)], y = 0, cos(xyTilt)) → f_'
  12. Stretch(Rotate[g, (-xyRotate)], y = 0, cos(xyTilt)) → g_'
  13. Textfield(height) → inputBox3
  14. Sequence(Segment[Element[value_f, k], Element[value_g, k]], k, 1, n) → base_{length}
  15. Stretch(Rotate[value_f, (-xyRotate)], y = 0, cos(xyTilt)) → base_f
  16. Stretch(Rotate[value_g, (-xyRotate)], y = 0, cos(xyTilt)) → base_g
  17. Sequence(Element[base_f, k] + (0, (sin(xyTilt) * height(Element[base_{length}, k]))), k, 1, n) → top_f
  18. Sequence(Element[base_g, k] + (0, (sin(xyTilt) * height(Element[base_{length}, k]))), k, 1, n) → top_g
  19. Sequence(Segment[Element[base_f, k], Element[top_f, k]], k, 1, n) → side_f
  20. Sequence(Segment[Element[base_g, k], Element[top_g, k]], k, 1, n) → side_g
  21. Sequence(Segment[Element[base_g, k], Element[base_f, k]], k, 1, n) → base
  22. Sequence(Segment[Element[top_g, k], Element[top_f, k]], k, 1, n) → top_{rectangle}
  23. Polygon(Element[top_f, n], Element[top_g, n], Element[base_g, n], Element[base_f, n]) → face_b, c, d, e, h
  24. Polygon(Element[top_f, 1], Element[top_g, 1], Element[base_g, 1], Element[base_f, 1]) → face_a, i, j, k, l
  25. Segment(C, D) → m
  26. Point(m) → E
  27. OrthogonalLine(E, m) → p
  28. Point(p) → F
  29. Segment(C, F) → q
  30. Segment(D, F) → r
  31. Sequence(Point[Element[base_g, k], Vector[(triangleRatio * Vector[Element[base_g, k], Element[base_f, k]])]], k, 1, n) → mids_{triangle}
  32. Sequence(Element[mids_{triangle}, k] + (0, ((Element[base_{length}, k] * triangleHeight) * sin(xyTilt))), k, 1, n) → top_{triangle}
  33. Sequence(Segment[Element[base_f, k], Element[top_{triangle}, k]], k, 1, n) → side_{ft}
  34. Sequence(Segment[Element[base_g, k], Element[top_{triangle}, k]], k, 1, n) → side_{gt}
  35. Polygon(Element[top_{triangle}, 1], Element[base_g, 1], Element[base_f, 1]) → face_{at}, s, t, a_1
  36. Polygon(Element[top_{triangle}, n], Element[base_g, n], Element[base_f, n]) → face_{bt}, b_1, c_1, d_1