Fundamental Theorem of Calculus
The theorem connecting differentiation and integration: the derivative of an integral recovers the integrand, and definite integrals equal differences of antiderivatives.
The fundamental theorem of calculus (FTC) is the bridge between the two halves of calculus: differentiation, which measures instantaneous rates of change, and integration, which accumulates quantities. It was discovered independently by Isaac Newton and Gottfried Leibniz in the 17th century, and it explains why the two operations are inverses of each other. Formally it has two parts.
Part 1 states that if f is continuous on [a, b], then the function F(x) = ∫ₐˣ f(t) dt — the accumulated area from a to x — is differentiable and its derivative is f(x) itself. In other words, integrating a function and then differentiating brings you back to the original function: the area function grows at exactly the rate the integrand dictates. This is why integration is described as the inverse of differentiation.
Part 2 is the computational workhorse. If F is any antiderivative of f (that is, F′ = f), then the definite integral from a to b equals F(b) − F(a). Without this result, computing areas required laborious limiting sums; with it, one simply finds an antiderivative and evaluates it at two points. For example, ∫₀¹ x² dx = 1/3 − 0 = 1/3, obtained instantly from the antiderivative x³/3.
∫ₐᵇ f(x) dx = F(b) − F(a), where F' = f
The FTC unifies seemingly different problems: areas under curves (integration) and tangent slopes (differentiation) are two views of one mathematical structure. Its reach extends everywhere calculus is used — physics (displacement as the integral of velocity, velocity as the derivative of position), probability (cumulative distribution functions as integrals of densities), engineering, and economics (marginal and total quantities). Part 2 also explains why integration by substitution and integration by parts work: they are the integral versions of the chain rule and the product rule.
Tags
calculus differentiation integration mathematics
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