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Volume Calculation Using the Disk-Washer Method

Volume Calculation Using the Disk-Washer Method
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2015-10-30 19:22
FunctionCalculusIntegral
In collections Calculus: Volumes of Solids Calculus: Volumes of Solids
This example demonstrates the process of calculating the volume of a solid of revolution using the disk-washer method. Two 2D views and a 3D perspective are provided, allowing for modifications to functions, rotation axis, and intervals. Commands used include Intersect, Min, Max, If, Textfield, IntegralBetween, Segment, Circle, Plane, etc. Suitable for calculus lessons on volume calculation.

GeoGebra Commands & Parameters

  1. Intersect(g, l_c, (1, 1)) → P_{gc}
  2. Intersect(f, l_c, (1, 2)) → P_{fc}
  3. Min(abs(rad_f), abs(rad_g)) → rad_i
  4. Max(abs(rad_f), abs(rad_g)) → rad_o
  5. If(rad_o ≠ rad_f, f_i, f_i) → rwash_{3out}
  6. Textfield(a) → inputBox_a
  7. Textfield(b) → inputBox_b
  8. Intersect(f, l_b, (3, 6)) → P_{fb}
  9. Intersect(g, l_b, (3, 1.7320508075688772)) → P_{gb}
  10. Intersect(f, l_a, (1, 2)) → P_{fa}
  11. Intersect(g, l_a, (1, 1)) → P_{ga}
  12. IntegralBetween(f, g, Min[x(P_{fa}), x(P_{ga})], Max[x(P_{fb}), x(P_{gb})]) → R
  13. Segment((0, y_{rotation}), (0, If[abs(rad_f) ≥ abs(rad_g), y(P_{fc}), y(P_{gc})])) → r_o
  14. Segment((0, y_{rotation}), (0, If[abs(rad_f) < abs(rad_g), y(P_{fc}), y(P_{gc})])) → r_i
  15. Circle(P, rad_i, y = 0) → cs3_i
  16. Circle(P, rad_o, y = 0) → cs3_o
  17. Plane((x_i, y_{rotation}, rad_o), x = x_i) → pln3d
  18. Textfield(n) → InputBox1
  19. If(rad_o ≟ rad_f, If[sgn(rad_f) ≥ 0, f_i(x) - y_{rotation}, y_{rotation} - f_i(x)], If[sgn(rad_g) ≥ 0, g_i(x) - y_{rotation}, y_{rotation} - g_i(x)]) → r_{outer}
  20. If(rad_o ≠ rad_f, If[sgn(rad_f) ≥ 0, f_i(x) - y_{rotation}, y_{rotation} - f_i(x)], If[sgn(rad_g) ≥ 0, g_i(x) - y_{rotation}, y_{rotation} - g_i(x)]) → r_{inner}
  21. IntegralBetween((π * r_{outer}(x)^(2)) - r_{inner}(x)^(2), x, a, b) → volume
  22. Circle(P, rad_i, y = 0) → cs3_{i2}
  23. SelectedIndex(lstTraceOpt) → lstTOIndex
  24. Intersect(rwash_{3out}, l_c, (1, 2)) → P_{oc}
  25. If(rad_o ≠ rad_f, f_i, g_i) → rwash_{3in}
  26. Intersect(rwash_{3in}, l_c, (1, 1)) → P_{ic}
  27. Segment(P_{oc}, P_{ic}) → r_{fg}
  28. Sequence(Segment[Rotate[P_{fc}, i°, l_r], Rotate[P_{gc}, i°, l_r]], i, 0, 360, 1) → wash_3
  29. Sequence(Rotate[P_{oc}, i°, l_r], i, 0, 360, 1) → wash_{3o}
  30. Sequence(Rotate[P_{ic}, i°, l_r], i, 0, 360, 1) → wash_{3i}
  31. Textfield(tf) → InputBox_f
  32. Textfield(tg) → InputBox_g