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Volumes by Cross Sections by Lisa Kramer

Volumes by Cross Sections by Lisa Kramer
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2018-03-20 18:12
Solid GeometryCalculusGeometric Construction
This example demonstrates how to calculate volumes using cross sections. Key mathematical concepts include solid geometry. Construction highlights the use of Plane, Segment, Intersect, Polygon, and other GeoGebra commands. Suitable for high school math education or students learning solid geometry.

GeoGebra Commands & Parameters

  1. Plane((0, 0, 0), (0, 1, 0), (0, 1, 1)) → s
  2. Plane((a, 0, 0), (a, 1, 1), (a, 1, 2)) → c
  3. Plane((b, 0, 0), (b, 1, 1), (b, 1, 2)) → d
  4. Segment((a, f(a)), (a, g(a))) → h
  5. Segment((b, f(b)), (b, g(b))) → i
  6. Intersect(f, h) → A
  7. Intersect(g, h, (-8, -6.968074739870145)) → B
  8. Intersect(f, i) → C
  9. Intersect(g, i, (8, -1.0319252601298547)) → D
  10. Polygon(A, B, (x(B), y(B), h), (x(A), y(A), h)) → q1, a_1, b_1, t, f_1
  11. Polygon(A, B, (a, (0.5 * (y(A) + y(B))), (sqrt(3) * h / 2))) → t1, g_1, a_2, b_2
  12. Midpoint(A, B) → E
  13. Sphere(E, A) → u
  14. Sequence(Polygon[(a + (i * deltax), g(a + (i * deltax)), 0), (a + (i * deltax), f(a + (i * deltax)), 0), 4, s], i, 0, n - 1, 1) → L3
  15. Polygon(A, B, (x(B), y(B), h)) → t2, k, a_3, b_3
  16. Sequence(Polygon[(a + (i * deltax), g(a + (i * deltax)), 0), (a + (i * deltax), f(a + (i * deltax)), 0), 3, s], i, 0, n - 1, 1) → L4
  17. Sequence(Polygon[(a + (i * deltax), f(a + (i * deltax)), 0), (a + (i * deltax), g(a + (i * deltax)), 0), (a + (i * deltax), g(a + (i * deltax)), Distance[(a + (i * deltax), f(a + (i * deltax)), 0), (a + (i * deltax), g(a + (i * deltax)), 0)])], i, 0, n - 1, 1) → L7
  18. Line((0, 0, 0), (1, 0, 0)) → m
  19. Line((0, 0, 0), (0, 1, 0)) → r
  20. Textfield(f) → InputBox1
  21. Textfield(g) → InputBox2
  22. Textfield(a) → InputBox3
  23. Textfield(b) → InputBox4
  24. Textfield(n) → InputBox5
  25. Sequence(Prism[Element[L3, i], deltax], i, 1, n, 1) → L2
  26. Sequence(Prism[Element[L4, i], deltax], i, 1, n, 1) → L5
  27. Sequence(Polygon[(a + (i * deltax), g(a + (i * deltax)), 0), (a + (i * deltax), f(a + (i * deltax)), 0), 5, s], i, 0, n - 1, 1) → L6
  28. Sequence(Prism[Element[L6, i], deltax], i, 1, n, 1) → L8
  29. Sequence(Midpoint[(a + (i * deltax), f(a + (i * deltax)), 0), (a + (i * deltax), g(a + (i * deltax)), 0)], i, 0, n - 1, 1) → L9
  30. Sequence((a + (i * deltax), f(a + (i * deltax)), 0), i, 0, n - 1, 1) → L10
  31. Sequence(Segment[Element[L9, i], Element[L10, i]], i, 1, n, 1) → radii
  32. Sequence((a + (i * deltax), g(a + (i * deltax)), 0), i, 0, n - 1, 1) → L11
  33. Sequence(Midpoint[Element[L10, i], Element[L11, i]], i, 1, n, 1) → L12
  34. Sequence(CircleArc[Element[L12, i], Element[L10, i], Element[L11, i], s], i, 1, n, 1) → L13
  35. Sequence(Prism[Element[L7, i], (-deltax)], i, 1, n, 1) → L