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The Definite Integral

Signed limit of Riemann sums · BC Calculus · greeting

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Owl · A skateboarder rides a wavy half-pipe profile drawn as y = f(x). From x = −2 to x = 2 the curve sits above the axis on the left half and dips below on the right. Someone claims the "area under the curve" on that interval is zero because the bumps cancel. Are they talking about geometric area, or about a signed quantity? The definite integral ∫ₐᵇ f is the limit of Riemann sums — and those sums carry the sign of f. Positive regions add; negative regions subtract. Why can that net "area" be zero even when the curve is nowhere flat?

Owl: A skateboarder rides a wavy half-pipe profile drawn as y = f(x). From x = −2 to x = 2 the curve sits above the axis on the left half and dips below on the right. Someone claims the "area under the curve" on that interval is zero because the bumps cancel. Are they talking about geometric area, or about a signed quantity? The definite integral ∫ₐᵇ f is the limit of Riemann sums — and those sums carry the sign of f. Positive regions add; negative regions subtract. Why can that net "area" be zero even when the curve is nowhere flat?

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