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Building Expressions with Riemann Sums (4)

Building Expressions with Riemann Sums (4)
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2021-03-09 21:45
CalculusIntegralGeometric ConstructionGraphing
This GeoGebra example demonstrates how to construct piecewise algebraic expressions and connect them to geometric approximations of a definite integral using upper and lower Riemann sums. It centers on expression-building with conditions (piecewise logic via If), arithmetic representation of fractions (FractionText), and interactive text/object display (Textfield) to label or parameterize the construction. The construction uses Polygon to create the geometric regions/blocks, then applies If to switch or define expressions according to interval logic, while Textfield and FractionText present the corresponding formulas clearly. It is implemented with many geometric points and polygon objects (e.g., pic1–pic10, poly0/poly3/poly7, along with function objects f, g, h, i) to support the visualization of sum-of-areas behavior toward the Riemann-sum limit. Suitable for upper secondary through first-year college learners, especially for calculus concepts (Riemann sums, definite integrals) and for practicing structured expression creation in GeoGebra.

GeoGebra Commands & Parameters

  1. Polygon(A0, B, 4) → poly0, f, g, h, i, A, C
  2. Polygon(A3, H, 4) → poly3, j, k, l, m, E, G
  3. Polygon(A7, P, 4) → poly7, n, p, q, r, I, K
  4. Polygon(A1, D, 4) → poly1, s, t, a, b, M, O
  5. Polygon(A6, N, 4) → poly6, c, d, e, f_1, Q, S
  6. Polygon(A2, H2, 4) → poly2, k_1, l_1, m_1, n_1, W, Z
  7. Polygon(A8, A8 + (2, 0), 4) → poly8, g_1, h_1, i_1, j_1, U, V
  8. Polygon(A9, T, 4) → poly9, p_1, q_1, r_1, s_1, A_1, B_1
  9. Polygon(A4, J, 4) → poly4, t_1, a_1, b_1, c_1, C_1, D_1
  10. Polygon(A5, L, 4) → poly5, d_1, e_1, f_2, g_2, E_1, F_1
  11. Polygon(K_1, L_1, 4) → poly10, l_2, m_2, n_2, p_2, M_1, N_1
  12. Polygon(O_1, P_1, 4) → poly12, q_2, r_2, s_2, t_2, Q_1, R_1
  13. Polygon(S_1, T_1, 4) → poly13, a_2, b_2, c_2, d_2, U_1, V_1
  14. Polygon(C_2, D_2, 4) → poly15, i_3, j_3, k_3, l_3, E_2, F_2
  15. Polygon(G_2, H_2, 4) → poly16, m_3, n_3, p_3, q_3, I_2, J_2
  16. Polygon(G_{3}, H_{3}, 4) → poly17, m_{4}, n_{4}, p_{4}, q_{4}, I_{3}, J_{3}
  17. Polygon(C_{3}, D_{3}, 4) → poly14, i_{4}, j_{4}, k_{4}, l_{4}, E_{3}, F_{3}
  18. FractionText(u) → text4
  19. If(aleft > aright, "Left side is bigger.", aleft < aright, "Right side is bigger", "Both sides are equal!") → text9
  20. Textfield(w) → InputBox1
  21. If(o_1 < 0, "less than", o_1 > 0, "greater than", o_1 ≟ 0, "equal to") → text12

AI Drawing Prompt Reference

Draw a collection of square grid blocks arranged horizontally, each representing an area segment for illustrating Riemann sums approximating a definite integral. Include several piecewise function curves defined by conditional logic, with different expressions on different intervals. Label each block with a fraction text showing the function value or area approximation. Add an input box and comparison texts that dynamically display whether the left or right side is larger, or whether a value is less than, greater than, or equal to zero. Arrange the blocks in a stepped or bar-chart layout to visualize upper and lower sums.
The prompt above was compiled from the original drawing commands and is for reference only.