Riemann Sums and the Definite Integral

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A Riemann sum estimates the area under a curve by splitting the interval into n thin rectangles and adding their areas, using left, right or midpoint heights. As n → ∞ the sum approaches the exact area, the definite integral ∫ₐᵇ f(x) dx.

The integral sign is a stretched S for sum, and dx reminds us that the rectangle widths shrink toward zero. Each rectangle has area f(xᵢ)·Δx, where Δx = (b − a)/n. Area above the x-axis counts as positive and area below counts as negative, so the integral gives net or signed area.

Example: to estimate the area under f(x) = x² from 0 to 2 with four rectangles of width 0.5 using right endpoints, add 0.5(0.25 + 1 + 2.25 + 4) = 3.75. The exact integral is 8/3, about 2.67, so the right sum overestimates for an increasing function; the left sum gives 1.75, an underestimate, and more rectangles improve both.

Integrals compute distance from a speed graph, total water used from a flow rate, work done by a force and probabilities from density curves. Tip: for an increasing function the left sum underestimates and the right sum overestimates, and the reverse holds for a decreasing function.