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数值积分的黎曼、梯形和辛普森方法

数值积分的黎曼、梯形和辛普森方法
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2019-12-02 18:39
函数数列微积分积分函数图像
本范例演示了数值积分中的黎曼、梯形和辛普森方法。通过构造几何图形，展示了这些方法的核心数学概念。使用了GeoGebra中的Point、LeftSum、TrapezoidalSum等命令，构建了黎曼和、梯形和、辛普森和等对象。适合高中数学教学和学习，特别是微积分和数值分析领域。

GeoGebra 命令及参数

  1. Point(xAxis) → B
  2. Point(xAxis) → A
  3. LeftSum(f, x(A), x(B), subintervals) → LeftRiemann
  4. LowerSum(f, x(A), x(B), subintervals) → LowerRiemann
  5. UpperSum(f, x(A), x(B), subintervals) → UpperRiemann
  6. TrapezoidalSum(f, x(A), x(B), subintervals) → TrapezoidRule
  7. Sequence((x(A) + (n * dx), 0), n, 0, subintervals) → bottoms
  8. Sequence((x(Element[bottoms, n]), f(x(Element[bottoms, n]))), n, 2, subintervals + 1) → topRight
  9. Sequence((x(Element[bottoms, n - 1]), f(x(Element[bottoms, n]))), n, 2, subintervals + 1) → topLeft
  10. Sequence(Polygon[Element[bottoms, n], Element[bottoms, n + 1], Element[topRight, n], Element[topLeft, n]], n, 1, subintervals) → RRRect
  11. Sum(Sequence[(Element[RRRect, n] * sgn(y(Element[topRight, n]))), n, 1, subintervals]) → RightRiemann
  12. Sequence((x(A) + (n * dx), f(x(A) + (n * dx))), n, 0, subintervals) → tops
  13. Sequence(FitPoly[{Element[tops, (2 * n) - 1], Element[tops, (2 * n)], Element[tops, (2 * n) + 1]}, 2], n, 1, subintervals / 2) → parabolas
  14. RectangleSum(f, x(A), x(B), subintervals, RiemannLeftRight) → VariableRiemann
  15. Sequence(Integral[Element[parabolas, n], x(Element[tops, (2 * n) - 1]), x(Element[tops, (2 * n) + 1])], n, 1, subintervals / 2) → ShowParabolas
  16. Textfield(f) → inputBox1