Integration & the Fundamental Theorem
Riemann sums, the definite integral, and the fundamental theorem of calculus.
For a bounded $f$ on $[a,b]$, partition the interval into $n$ pieces; pick sample points $x_i^*$ in each. The Riemann sum
$$S_n = \sum_{i=1}^n f(x_i^*)\, \Delta x_i$$approaches the definite integral $\int_a^b f(x)\, dx$ as the mesh size $\to 0$ (provided the limit is independent of partition and sample points — i.e., $f$ is Riemann integrable). Continuous functions on closed intervals always qualify.
The fundamental theorem of calculus:
$$\frac{d}{dx} \int_a^x f(t)\, dt = f(x), \qquad \int_a^b f'(x)\, dx = f(b) - f(a).$$It ties integration (geometry: area) to differentiation (rate of change). Techniques: substitution ($\int f(g(x)) g'(x) dx = \int f(u) du$), integration by parts ($\int u\,dv = uv - \int v\,du$), and partial fractions.