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Goblet Volume via Rotation and Integration

Goblet Volume via Rotation and Integration
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2017-05-18 01:59
Solid GeometryCalculusIntegral
This example demonstrates the generation of a goblet 3D model from a 2D shape through rotation and the calculation of its volume using definite integrals. Key math concepts include definite integral, integral calculus, rotation, volume, equilateral triangles, subtraction, bar chart, means, and equations. It utilizes GeoGebra commands like Point, Segment, Sequence, Function, Mirror, Textfield, Integral, Translate, Corner, Element, CurveCartesian, Circle, Intersect, Polygon. Suitable for high school students and math enthusiasts learning about rotational volumes and integral applications.

GeoGebra Commands & Parameters

  1. Point(offsetYaxis) → axis
  2. Point(offsetXaxis) → a_f
  3. Point(offsetXaxis) → b_f
  4. Segment(a_f + (0, sliderXoffset2), b_f + (0, sliderXoffset2)) → innerslider
  5. Point(innerslider) → a_g
  6. Point(innerslider) → b_g
  7. Sequence((af_x + i, axis_y), i, 0, (Δx * (n - 1)), Δx) → list_{centers}
  8. Sequence((af_x + i + (leftright * Δx), f(af_x + i + (leftright * Δx))), i, 0, (Δx * (n - 1)), Δx) → listf_{points}
  9. Sequence(If[(x(Element[list_{centers}, i]) < ag_x) ∨ (x(Element[list_{centers}, i]) > bg_x - Δx), Corner[1] + (-1, -1), (x(Element[listf_{points}, i]), g(x(Element[listf_{points}, i])))], i, 1, n) → listg_{points}
  10. Function(f, af_x, bf_x) → f_1
  11. Sequence(abs(y(Element[listf_{points}, i]) - y(Element[list_{centers}, i])), i, 1, n) → listf_{radii}
  12. Sequence((0, (Element[listf_{radii}, i] * sqrt(1 - tilt^(2)))), i, 1, n) → listf_c
  13. Sequence(Ellipse[Element[listf_{foci1}, i], Element[listf_{foci2}, i], Element[listf_{radii}, i]], i, 1, n) → listf_{ellipses1}
  14. Sequence(Segment[Element[listf_{points}, i] - ((leftright * Δx), 0), Element[listf_{points}, i] + (((1 - leftright) * Δx), 0)], i, 1, n) → listf_{segments1}
  15. Mirror(listf_{segments1}, axisline) → listf_{segments2}
  16. Textfield(f) → inputBox1
  17. Integral((π * (f(x) - axis_y)^(2)), af_x, bf_x) → Vf_{actual}
  18. Segment(a_f, (af_x, axis_y)) → segment_a
  19. Segment(b_f, (bf_x, axis_y)) → segment_b
  20. Sequence(If[(x(Element[list_{centers}, i]) < ag_x) ∨ (x(Element[list_{centers}, i]) > bg_x - Δx), 0, abs(y(Element[listg_{points}, i]) - y(Element[list_{centers}, i]))], i, 1, n) → listg_{radii}
  21. Function(g, ag_x, bg_x) → g_1
  22. Sequence((0, (Element[listg_{radii}, i] * sqrt(1 - tilt^(2)))), i, 1, n) → listg_c
  23. Sequence(Ellipse[Element[listg_{foci1}, i], Element[listg_{foci2}, i], Element[listg_{radii}, i]], i, 1, n) → listg_{ellipses1}
  24. Textfield(g) → inputBox2
  25. Translate(listf_{ellipses1}, Vector[(0, 0), (Δx, 0)]) → listf_{ellipses2}
  26. Translate(listg_{ellipses1}, Vector[(0, 0), (Δx, 0)]) → listg_{ellipses2}
  27. Sequence(Segment[Element[listg_{points}, i] - ((leftright * Δx), 0), Element[listg_{points}, i] + (((1 - leftright) * Δx), 0)], i, 1, n) → listg_{segments1}
  28. Mirror(listg_{segments1}, axisline) → listg_{segments2}
  29. Integral((π * (g(x) - axis_y)^(2)), ag_x, bg_x) → Vg_{actual}
  30. Corner(6) → A
  31. Segment(a_g, (ag_x, axis_y)) → segmenti_a
  32. Segment(b_g, (bg_x, axis_y)) → segmenti_b
  33. Segment((x(a_f), sliderXoffset3), (x(b_f), sliderXoffset3)) → ellipseslider
  34. Point(ellipseslider) → highlight
  35. Element(list_{centers}, ellipsenumber) → ellipsecenter
  36. Element(listf_{radii}, ellipsenumber) → ellipsef_a
  37. Element(listf_{ellipses2}, ellipsenumber) → ellipsef
  38. CurveCartesian(x(ellipsecenter) + (ellipsef_b * cos(t)), y(ellipsecenter) + (ellipsef_a * sin(t)), t, 1.5707963267948966, 4.71238898038469) → ellipsehalf1_f
  39. Element(listf_{segments1}, ellipsenumber) → segmentf1
  40. Element(listf_{segments2}, ellipsenumber) → segmentf2
  41. Element(listg_{ellipses2}, ellipsenumber) → ellipseg
  42. Element(listg_{radii}, ellipsenumber) → ellipseg_a
  43. Circle((0, 0), ellipseg_a) → d
  44. Circle((Δx / tilt, 0), ellipseg_a) → e
  45. Intersect(d, e) → B, C, ,
  46. CurveCartesian(x(ellipsecenter) + (ellipseg_b * cos(t)), y(ellipsecenter) + (ellipseg_a * sin(t)), t, (-β), β) → ellipsepart_g
  47. CurveCartesian(x(ellipsecenter) + Δx + (ellipsef_b * cos(t)), y(ellipsecenter) + (ellipsef_a * sin(t)), t, 1.5708, 4.71238898038469) → ellipsehalf2_f
  48. CurveCartesian(x(ellipsecenter) + Δx + (ellipsef_b * cos(t)), y(ellipsecenter) + (ellipsef_a * sin(t)), t, -1.5708, 1.5707963267948966) → ellipsehalf3_f
  49. Polygon({(0, ellipsef_a), (Δx, ellipsef_a), (Δx, (-ellipsef_a)), (0, (-ellipsef_a))} + ellipsecenter) → poly1
  50. Function((-f(x)) + (2 * axis_y), af_x, bf_x) → f_2
  51. Function((-g(x)) + (2 * axis_y), ag_x, bg_x) → g_2
  52. CurveCartesian(x(ellipsecenter) + Δx + (ellipseg_b * cos(t)), y(ellipsecenter) + (ellipseg_a * sin(t)), t, π - β, π + β) → ellipsepartb_g