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Volume of Solid with Semi-Circular Cross Sections

Volume of Solid with Semi-Circular Cross Sections
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2015-11-02 06:07
CalculusIntegralGeometric Construction
In collections Calculus: Volumes of Solids Calculus: Volumes of Solids
This example helps visualize a solid built upon semi-circular cross sections. By adjusting parameters such as rectangle height, functions, limits, and axis of rotation, one can observe the formation of the solid. Key mathematical concepts include integration and volume of solids of revolution. GeoGebra commands used include Intersect, Textfield, Min, Max, IntegralBetween, etc. Suitable for high school and above students learning volume calculation of solids of revolution in calculus.

GeoGebra Commands & Parameters

  1. Intersect(g, l_c, (1, 1)) → P_{gc1}
  2. Intersect(f, l_c, (1, 2)) → P_{fc1}
  3. Textfield(a) → inputBox_a
  4. Textfield(b) → inputBox_b
  5. Min(abs(rad_f), abs(rad_g)) → rad_i
  6. Max(abs(rad_f), abs(rad_g)) → rad_o
  7. Intersect(f, l_b, (5, 10)) → P_{fb}
  8. Intersect(g, l_b, (5, 2.23606797749979)) → P_{gb}
  9. Intersect(f, l_a, (1, 2)) → P_{fa}
  10. Intersect(g, l_a, (1, 1)) → P_{ga}
  11. IntegralBetween(f, g, Min[x(P_{fa}), x(P_{ga})], Max[x(P_{fb}), x(P_{gb})]) → R
  12. Segment((0, y_{rotation}), (0, If[abs(rad_f) ≥ abs(rad_g), y(P_{fc1}), y(P_{gc1})])) → r_o
  13. Segment((0, y_{rotation}), (0, If[abs(rad_f) < abs(rad_g), y(P_{fc1}), y(P_{gc1})])) → r_i
  14. Plane((x_i, y_{rotation}, rad_o), x = x_i) → pln3d
  15. Textfield(n) → InputBox1
  16. If(x(P_{gc2}) < x(P_{fc2}), Semicircle[P_{gc2}, P_{fc2}], Semicircle[P_{fc2}, P_{gc2}]) → semic_2
  17. If(y(P_{gc1}) < y(P_{fc1}), CircumcircleArc[P_{gc3}, P_{sc3}, P_{fc3}], CircumcircleArc[P_{fc3}, P_{sc3}, P_{gc3}]) → semic_3
  18. If(y(P_{gc1}) < y(P_{fc1}), CircumcircleArc[P_{gc1}, P_{sc1}, P_{fc1}], CircumcircleArc[P_{fc1}, P_{sc1}, P_{gc1}]) → semic_1
  19. SelectedIndex(lstTraceOpt) → lstTOIndex
  20. Textfield(tf) → InputBox_f
  21. Textfield(tg) → InputBox_g

AI Drawing Prompt Reference

Draw a solid with semi-circular cross sections. Define two functions f and g, and generate semi-circular cross sections based on these functions within a specified interval. Adjust the function parameters and interval to observe the formation of the solid. Use integration to calculate the volume of the solid.
The prompt above was compiled from the original drawing commands and is for reference only.