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Calculus: Riemann Sums vs. Trapezoidal Rule for Numerical Integration

Calculus: Riemann Sums vs. Trapezoidal Rule for Numerical Integration
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2015-10-14 22:17
FunctionSequenceCalculusIntegralGraphing
This example demonstrates the comparison between Riemann sums and the Trapezoidal Rule for approximating definite integrals. Key concepts involve numerical integration, limits, and area under curves. Construction utilizes GeoGebra commands like RectangleSum, TrapezoidalSum, Sequence, and Integral to visualize subdivisions and calculate areas. Ideal for calculus students to understand the convergence of approximation methods to the exact integral value.

GeoGebra Commands & Parameters

  1. RectangleSum(f_1, a, b, Subdivisions, int) → e
  2. Textfield(f_1) → InputBox1
  3. Textfield(a) → InputBox2
  4. Textfield(c) → InputBox4
  5. Integral(f_1, a, b) → d
  6. Segment(A, C) → Δx
  7. Sequence(a + (dx * int) + (dx * i), i, 0, Subdivisions) → list1
  8. Sequence((Element[list1, i], 0), i, 1, Subdivisions) → list2
  9. Sequence((Element[list1, i], f_1(Element[list1, i])), i, 1, Subdivisions) → list3
  10. Sequence(Segment[Element[list2, i], Element[list3, i]], i, 1, Subdivisions) → list4
  11. Textfield(b) → InputBox3
  12. TrapezoidalSum(f_1, a, b, Subdivisions) → f
  13. RectangleSum(f_1, a, b, Subdivisions, 0) → left
  14. RectangleSum(f_1, a, b, Subdivisions, 1) → right
  15. RectangleSum(f_1, a, b, Subdivisions, 0.5) → midpoint
  16. TrapezoidalSum(f_1, a, b, Subdivisions) → trapezoid