Riemann sums become the integral
This animation approximates the area under y = x^2 from 0 to 2 using rectangles, starting with four and doubling to eight, sixteen, and thirty-two. As rectangle count grows, the staircase shape hugs the curve more closely and the total area converges toward the exact value 8/3. It visually demonstrates that a definite integral is the limit of Riemann sums as rectangle width shrinks toward zero, useful for introducing integration concepts.
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The prompt that made it
Approximating the area under y = x^2 from 0 to 2 with rectangles. Four rectangles, then eight, sixteen and thirty-two: the staircase hugs the curve and the total area settles toward the exact integral, 8/3. The point: an integral is the limit of ever-finer rectangle sums.
Make the next one in this series
- Compare left, right, and midpoint Riemann sums for the same function and interval
- Show the same convergence process using trapezoids instead of rectangles
- Visualize the error between Riemann sum approximations and the true integral as rectangles increase
