Riemann Sums & 3D Integrals

Riemann Sums & 3D Integrals turns multivariable integration into a tactile 3D geometry model. Students see a translucent curved surface z = f(x, y) hovering over a subdivided xy-domain filled with height-colored rectangular columns. Each column has base area Δx·Δy and height f(xᵢ*, yⱼ*) sampled according to the selected rule. Dragging the glowing probe across the surface highlights the active prism and displays its individual volume dV = f(x, y) Δx Δy, while dragging the corner boundary handle resizes the integration domain. By sliding the grid resolution N from coarse blocks up to a 20×20 mesh of 400 columns across Paraboloid Dome, Saddle Wave, and Gaussian Hill surfaces, learners observe how the double sum converges to the exact analytical double integral.

• Set Grid subdivisions N to 4 and switch Sample rule between Lower Corner, Upper Corner, and Midpoint to compare how each rule under- or over-estimates the true volume.
• Slowly increase N from 2 up to 20 and watch the Approximation error percentage approach zero as Total columns N² reaches 400.
• Drag the glowing red sample probe across the surface or click individual columns to inspect their local height z and differential volume dV.
• Drag the gold corner handle on the base plane to expand or shrink the square integration domain [-B, B]² and observe how both volumes scale.
• Switch Surface function between Paraboloid Dome, Saddle Wave, and Gaussian Hill to test how surface curvature affects midpoint accuracy.

1. 1Interpret a double integral ∬ f(x, y) dA geometrically as the net volume underneath a 3D surface z = f(x, y).
2. 2Explain how partitioning a 2D domain into N² sub-rectangles produces a Riemann sum Σ Σ f(xᵢ*, yⱼ*) Δx Δy.
3. 3Compare the accuracy of Midpoint, Lower Corner, and Upper Corner sampling rules for a given grid resolution N.
4. 4Describe why approximation error approaches zero in the limit as Δx and Δy approach zero.

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