Riemann Sums Revisited

Calculus 2 · Axiom Academy

Approximating a definite integral by tiling the area under a curve with rectangles — left, right, and midpoint — and watching the estimate sharpen as the count grows. 1. Rectangles That Tile the Area Take on the interval [1, 4] . We want the shaded area . Cut [1,4] into n strips of equal width , stand a rectangle on each strip, and use the left endpoint of the strip for its height. The sum of those rectangle areas is the left Riemann sum L_n — a first estimate of the true area. Each rectangle: width , height f at the left edge The only thing that changes between the three classic rules is where on each strip you read the height . Watch the same four rectangles re-anchor from the left edge, to the middle, to the right edge of every strip. Height from the left endpoint x_ i-1 . For a rising curve, tops fall short — an under-estimate. Height from the midpoint. The top cuts the curve, so the slice's over- and under-shoot cancel. Height from the right endpoint x_i . For a rising curve, tops overshoot — an over-estimate. Hold the method fixed (midpoint) and let n grow: . Each refinement halves the strip width, the staircase tightens onto the curve, and the leftover gap between the rectangles and the true area collapses. "Close" isn't good enough — we want a guarantee. The slope and curvature of f cap the error. Watch the two error curves drop as n grows: the left/right error falls like 1/n , but the midpoint error dives like 1/n^2 .

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