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绕x轴旋转体

绕x轴旋转体
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2019-11-16 02:15
立体几何微积分积分几何作图
①本范例演示了平面区域绕x轴旋转形成的旋转体；②涉及的核心数学概念包括微积分中的积分计算、旋转几何、体积计算；③作图要点使用了Textfield, CircleSector, Surface, Circle, CurveCartesian等GeoGebra命令和对象；④适合高中及以上学生学习和理解立体几何与微积分应用。

GeoGebra 命令及参数

  1. Textfield(a) → inputa
  2. Textfield(b) → b1
  3. Textfield(f1) → inputf
  4. CircleSector((a, 0, 0), (a, f(a), 0), (a, 0, f(a)), xOyPlane) → sectorA11
  5. CircleSector((a, 0, 0), (a, 0, f(a)), (a, (-f(a)), 0), xOyPlane) → sectorA12
  6. CircleSector((b, 0, 0), (b, f(b), 0), (b, (f(b) * cos(umax)), (f(b) * sin(umax))), xOyPlane) → sectorB1
  7. Surface(t, (f(t) * cos(u)), (f(t) * sin(u)), t, a, b, u, 0, umax) → surf1
  8. Circle((cs, 0, 0), f(cs), x = 0) → crossSect
  9. CurveCartesian(t, (f(t) * cos(umax)), (f(t) * sin(umax)), t, a, b) → fcurve
  10. Surface(t2, ((u2 * f(t2)) * cos(umax)), ((u2 * f(t2)) * sin(umax)), t2, a, b, u2, 0, 1) → surfcap
  11. CircleSector((b, 0, 0), (b, f(b), 0), (b, (cos((5 * π / 6)) * f(b)), (sin((5 * π / 6)) * f(b))), xOyPlane) → sectorB11
  12. CircleSector((b, 0, 0), (b, (cos((2 * π / 3)) * f(b)), (sin((2 * π / 3)) * f(b))), (b, (cos((4 * π / 3)) * f(b)), (sin((4 * π / 3)) * f(b))), xOyPlane) → sectorB12
  13. CircleSector((b, 0, 0), (b, (cos((7 * π / 6)) * f(b)), (sin((7 * π / 6)) * f(b))), (b, (cos(π / 12) * f(b)), (sin(π / 12) * f(b))), xOyPlane) → sectorB13
  14. CircleSector((b, 0, 0), (b, (cos((2 * π / 3)) * f(b)), (sin((2 * π / 3)) * f(b))), (b, (f(b) * cos(umax)), (f(b) * sin(umax))), xOyPlane) → sectorB2
  15. CircleSector((b, 0, 0), (b, (cos((7 * π / 6)) * f(b)), (sin((7 * π / 6)) * f(b))), (b, (f(b) * cos(umax)), (f(b) * sin(umax))), xOyPlane) → sectorB3
  16. Circle((b, 0, 0), f(b), yzplane) → capb
  17. CircleSector((a, 0, 0), (a, f(a), 0), (a, (f(a) * cos(umax)), (f(a) * sin(umax))), xOyPlane) → sectorA1
  18. CircleSector((a, 0, 0), (a, (cos((2 * π / 3)) * f(a)), (sin((2 * π / 3)) * f(a))), (a, (f(a) * cos(umax)), (f(a) * sin(umax))), xOyPlane) → sectorA2
  19. CircleSector((a, 0, 0), (a, (cos((7 * π / 6)) * f(a)), (sin((7 * π / 6)) * f(a))), (a, (f(a) * cos(umax)), (f(a) * sin(umax))), xOyPlane) → sectorA3
  20. Circle((a, 0, 0), f(a), yzplane) → capa