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Riemann, Trapezoid, and Simpson's Methods for Numerical Integration

Riemann, Trapezoid, and Simpson's Methods for Numerical Integration
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2019-12-02 18:39
FunctionSequenceCalculusIntegralGraphing
This example demonstrates numerical integration methods including Riemann, Trapezoid, and Simpson's methods. It constructs geometric figures to illustrate the core mathematical concepts. GeoGebra commands such as Point, LeftSum, TrapezoidalSum, and RectangleSum are used to create Riemann sums, trapezoidal sums, and Simpson's sums. Suitable for high school mathematics education, particularly in calculus and numerical analysis.

GeoGebra Commands & Parameters

  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