Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links differentiation and integration, showing they are inverse operations and providing a way to evaluate definite integrals.
Formula
\int_a^b f(x)\, dx = F(b) - F(a)
Definition
The Fundamental Theorem of Calculus is the big discovery connecting two ideas that seem different: derivatives (rates of change) and integrals (areas), showing that integration and differentiation are opposites, like multiplication and division. Part 1: if $f$ is continuous on $[a,b]$, then $G(x) = \int_a^x f(t)\, dt$ is differentiable and $G'(x) = f(x)$. Part 2: if $F$ is any antiderivative of $f$, then $\int_a^b f(x)\, dx = F(b) - F(a)$. The theorem generalizes far beyond one dimension via Stokes' theorem, $\iint_M d\omega = \int_{\partial M} \omega$, which unifies the FTC, Green's theorem, the divergence theorem, and the classical Stokes' theorem, showing that the derivative $d$ and the boundary operator $\partial$ are adjoint operators.
Example
To find $\int_1^4 2x\, dx$, find the antiderivative $F(x) = x^2$, then subtract: $F(4) - F(1) = 16 - 1 = 15$, no rectangles needed. Part 1 in action: $d/dx[\int_0^x \sin(t)\, dt] = \sin(x)$. Part 2 in action: $\int_0^{\pi/2} \cos(x)\, dx = \sin(\pi/2) - \sin(0) = 1$. In one dimension, Stokes' theorem reduces to $\int_a^b F'(x)\, dx = F(b) - F(a)$, the FTC itself.
Key Insight
This theorem is why you can skip drawing rectangles and just use antiderivatives, the bridge between the two halves of calculus. Part 1 shows that every continuous function has an antiderivative, even if you cannot write it with elementary functions, while Part 2 makes calculation practical. The FTC is the one-dimensional case of the most profound theorem in differential geometry, and every "by parts" or "change of variables" formula in multivariable calculus traces back to this duality between $d$ and $\partial$.