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Volumes by Cross Section

Volumes by Cross Section
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2017-04-20 04:42
FunctionSolid GeometryCalculusIntegralGeometric Construction
This example demonstrates the calculation of volumes using the method of cross sections, involving core mathematical concepts such as volume, ratios, constructions, tangent function, stochastic process or random process, and reflection. Key construction points include the use of GeoGebra commands like Sequence, Stretch, Line, Textfield, Polygon, Segment, Point, and OrthogonalLine, with objects like 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. Suitable for high school and above students to learn and understand volume calculation methods.

GeoGebra Commands & Parameters

  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