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Volumes of Intersecting Cylinders

Volumes of Intersecting Cylinders
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2015-07-16 22:46
Solid Geometry
In collections intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve Volumes in Calculus Volumes in Calculus
This example demonstrates the volumes of intersecting cylinders with a radius of 2. Key mathematical concepts include the volume calculation in solid geometry. Using GeoGebra commands such as Cylinder, IntersectPath, Point, etc., the intersecting cylinders can be constructed. Suitable for high school or university solid geometry education.

GeoGebra Commands & Parameters

  1. Cylinder((4, 0, 0), (-4, 0, 0), 2) → a, c, d, b
  2. Cylinder((0, -4, 0), (0, 4, 0), 2) → e, f, g, h
  3. IntersectPath(plane1, e) → k
  4. Point(k) → A
  5. Conic((2, 2, 0), (-2, -2, 0), (0, 0, 2), (0, 0, -2), (1.4142135623730951, 1.4142135623730951, 1.4142135623730951)) → p
  6. IntersectPath(plane2, a) → q
  7. CurveCartesian((2 * cos(t)), (2 * cos(t)), (2 * sin(t)), t, 0, 6.28319) → crv1
  8. CurveCartesian((2 * cos(t)), (-2 * cos(t)), (2 * sin(t)), t, 0, 6.28319) → crv2
  9. Surface(u, v / ((-u) ≤ v ≤ u), sqrt(4 - u^(2)), u, -2, 2, v, -2, 2) → SecTop1
  10. Surface((-u), v / ((-u) ≤ v ≤ u), sqrt(4 - u^(2)), u, -2, 2, v, -2, 2) → SecTop3
  11. Surface(u / ((-v) ≤ u ≤ v), (-v), sqrt(4 - v^(2)), u, -2, 2, v, -2, 2) → SecTop2
  12. Surface(u / ((-v) ≤ u ≤ v), v, sqrt(4 - v^(2)), u, -2, 2, v, -2, 2) → SecTop4
  13. Surface(u, v / ((-u) ≤ v ≤ u), (-sqrt(4 - u^(2))), u, -2, 2, v, -2, 2) → SecBottom1
  14. Surface(u, (-u), v / ((-sqrt(4 - u^(2))) ≤ v ≤ sqrt(4 - u^(2))), u, 0, 2, v, -2, 2) → SectSide1_1
  15. Surface(u, u, v / ((-sqrt(4 - u^(2))) ≤ v ≤ sqrt(4 - u^(2))), u, 0, 2, v, -2, 2) → SectSide1_2
  16. Surface((-u), u, v / ((-sqrt(4 - u^(2))) ≤ v ≤ sqrt(4 - u^(2))), u, 0, 2, v, -2, 2) → SecSide3_1
  17. Surface(u / ((-v) ≤ u ≤ v), (-v), (-sqrt(4 - v^(2))), u, -2, 2, v, -2, 2) → SecBottom2
  18. Surface(u, u, v / ((-sqrt(4 - u^(2))) ≤ v ≤ sqrt(4 - u^(2))), u, -2, 0, v, -2, 2) → SecSide2_1
  19. Surface((-u), v / ((-u) ≤ v ≤ u), (-sqrt(4 - u^(2))), u, -2, 2, v, -2, 2) → SecBottom3
  20. Surface(u / ((-v) ≤ u ≤ v), v, (-sqrt(4 - v^(2))), u, -2, 2, v, -2, 2) → SecBottom4
  21. Polygon((ss, ss, PlaneHeight), ((-ss), ss, PlaneHeight), ((-ss), (-ss), PlaneHeight), (ss, (-ss), PlaneHeight)) → Square, i, j, l, m