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Visualization of Riemann Sum in 3D Space

Visualization of Riemann Sum in 3D Space
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2015-06-21 02:22
Solid GeometryCalculusIntegralGraphing
In collections surface 3D /3D-Oberflächen 1 surface 3D /3D-Oberflächen 1
This example visualizes the Riemann sum approximation for volume under a surface in 3D space. Core concepts include multiple integrals, Riemann sums, and solid geometry volume estimation. Construction utilizes GeoGebra commands like Sequence for grid generation, Polygon for prism building, and Point/Min/Max for boundary definition. Suitable for high school math extension or university calculus courses to intuitively understand the geometric meaning of double integrals.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Point(xAxis) → B
  3. Min(x(B), x(A)) → xmin
  4. Max(x(B), x(A)) → xmax
  5. Point(yAxis) → C
  6. Point(yAxis) → D
  7. Max(y(C), y(D)) → ymax
  8. Min(y(C), y(D)) → ymin
  9. Sequence(Sequence[(xmin + (dx * i), ymin + (dy * j)), i, 0, m], j, 0, n) → lista3
  10. Polygon((xmin, ymin), (xmax, ymin), (xmax, ymax), (xmin, ymax)) → R, a, b, c, d
  11. Sequence(Sequence[(xmin + (dx * i), ymin + (dy * j), f(xmin + (dx * i), ymin + (dy * j))), i, 0, m], j, 0, n) → Imagens
  12. Sequence(Sequence[Polygon[Element[Element[lista3, j], i], Element[Element[lista3, j], i + 1], Element[Element[lista3, j + 1], i + 1], Element[Element[lista3, j + 1], i]], i, 1, m], j, 1, n) → Part
  13. Sequence(Sequence[Prism[Element[Element[Part, i], j], Element[Element[Imagens, i], j]], j, 1, m], i, 1, n) → Prismas
  14. Sequence((xmin + (dx * i), 0), i, 0, m) → lista1
  15. Sequence((0, ymin + (dy * i)), i, 0, n) → lista2
  16. Textfield(f) → ct1

AI Drawing Prompt Reference

Visualize the Riemann sum approximation for the volume under a surface in 3D space. Start by defining two points on the x-axis and y-axis, and determine their minimum and maximum values to create a rectangular grid. Then, generate multiple quadrilaterals based on the function's values at grid points, each representing a small volume element. Stack prisms for each quadrilateral along its base to approximate the volume under the surface. Finally, label the function expression to illustrate its relationship with the geometric shape.
The prompt above was compiled from the original drawing commands and is for reference only.