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Calculus: Fundamental Theorems & Definite Integrals

Calculus formalizes continuous rates of change and quantitative area accumulation under functions via the Fundamental Theorem of Calculus:

\frac{d}{dx} \left[ \int_{a}^{x} f(t)\,dt \right] = f(x)

Let $F^{'} (x) = f (x)$ . The net accumulation over any closed interval $[a, b]$ evaluates to the upper bound minus lower bound antiderivative:

\int_{a}^{b} f(x)\,dx = F(b) - F(a)

Derived directly from the differential product rule $d(uv) = u\,dv + v\,du$ :

\int u \, dv = u v - \int v \, du