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圆环法体积计算演示

圆环法体积计算演示
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2019-04-04 09:29
函数立体几何微积分积分几何作图
①本范例演示通过圆环法计算旋转体体积的过程；②涉及核心数学概念包括积分、旋转体体积、函数图像；③构造过程中运用了Sequence（数列）、Polygon（多边形）、Rotate（旋转）、Integral（积分）、RectangleSum（矩形求和）、Surface（曲面）等GeoGebra命令；④适合高中及以上学生学习和理解微积分在几何体积计算中的应用。

GeoGebra 命令及参数

  1. Sequence(Polygon[((k * h), f(((k + p) * h))), (((k + 1) * h), f(((k + p) * h))), (((k + 1) * h), g(((k + p) * h))), ((k * h), g(((k + p) * h)))], k, 0, m - 1) → Rectangles
  2. Polygon(((DK * h), g(((DK + p) * h))), (((DK + 1) * h), g(((DK + p) * h))), (((DK + 1) * h), f(((DK + p) * h))), ((DK * h), f(((DK + p) * h)))) → Rectangle1, b, c, d, e
  3. Rotate(Rectangle1, α, a) → Rectangle1'
  4. Integral((π * ((4 - f(x))^(2) - (4 - g(x))^(2))), 0, 2) → VolInteg
  5. Rotate(f_1, α, a) → i
  6. Rotate(g_1, α, a) → j
  7. Rotate(Rectangles, α, a) → Rectangles'
  8. RectangleSum(A, 0, (m * h), m, p) → WS
  9. Surface(u, -2 - ((-2 - g(u)) * cos(β)), ((-2 - g(u)) * sin(β)), u, 0, 2, β, 0, (2 * π)) → Inner
  10. Surface(u, -2 - ((-2 - g(((DK + p) * h))) * cos(β)), ((-2 - g(((DK + p) * h))) * sin(β)), u, (DK * h), ((DK + 1) * h), β, 0, α) → Inner1
  11. Surface(u, -2 - ((-2 - f(u)) * cos(β)), ((-2 - f(u)) * sin(β)), u, 0, 2, β, 0, (2 * π)) → Outer
  12. Surface(u, -2 - ((-2 - f(((DK + p) * h))) * cos(β)), ((-2 - f(((DK + p) * h))) * sin(β)), u, (DK * h), ((DK + 1) * h), β, 0, α) → Outer1
  13. Sequence(Surface[u, -2 - ((-2 - g(((k + p) * h))) * cos(β)), ((-2 - g(((k + p) * h))) * sin(β)), u, (k * h), ((k + 1) * h), β, 0, α], k, 0, m - 1) → Inners
  14. Sequence(Surface[u, -2 - ((-2 - f(((k + p) * h))) * cos(β)), ((-2 - f(((k + p) * h))) * sin(β)), u, (k * h), ((k + 1) * h), β, 0, α], k, 0, m - 1) → Outers
  15. Sequence(Surface[(k * h), -2 - ((-2 - u) * cos(β)), ((-2 - u) * sin(β)), u, g(((k + p) * h)), f(((k + p) * h)), β, 0, α], k, 0, m - 1) → SidesLeft
  16. Sequence(Surface[((k + 1) * h), -2 - ((-2 - u) * cos(β)), ((-2 - u) * sin(β)), u, g(((k + p) * h)), f(((k + p) * h)), β, 0, α], k, 0, m - 1) → SidesRight
  17. Surface((DK * h), -2 - ((-2 - u) * cos(β)), ((-2 - u) * sin(β)), u, g(((DK + p) * h)), f(((DK + p) * h)), β, 0, α) → SideLeft1
  18. Surface((DK * h), -2 - ((-2 - u) * cos(β)), ((-2 - u) * sin(β)), u, g(((DK + p) * h)), f(((DK + p) * h)), β, 0, α) → SideRight1