Definite integral

$\int_a^b f(x)\,\dd x$ is the limit of Riemann sums: the signed area between the graph and the axis, or, more generally, the total accumulated by a rate $f$ over $[a, b]$.

Many quantities are "rate × amount" only when the rate is constant: distance = speed × time, mass = density × volume, work = force × distance. When the rate varies, chop the domain into pieces where it is nearly constant, add up, and take the limit. The integral is that procedure, and it solves the problem of accumulating a varying quantity.

Geometrically: area under the curve, counting area below the axis as negative. Statistically: $\frac{1}{b-a}\int_a^b f$ is the average value of $f$ on $[a,b]$ — which is why an integral can be estimated by averaging $f$ at random points (Monte Carlo). Physically: the total change produced by a rate.

$f$ is Riemann integrable on $[a,b]$ if the Riemann sums $\sum f(x_i^\ast)\Delta x_i$ converge to the same number $I$ for every choice of sample points as the mesh $\max\Delta x_i \to 0$; then $\int_a^b f = I$. Every continuous function (and every bounded function with finitely many discontinuities) is integrable. Properties: linearity, additivity over intervals, monotonicity ($f \le g \Rightarrow \int f \le \int g$).

Exactly, when an antiderivative is known: Barrow's rule $F(b) - F(a)$ (fundamental theorem). Otherwise numerically: Riemann/trapezoid/Simpson/Gaussian quadrature in low dimension, Monte Carlo in high dimension.

Energy of a signal $x(t) = \sin(2\pi t)$ over one second: $\int_0^1 \sin^2(2\pi t)\,\dd t = \frac12$. The same integral, as a sum over samples $\frac1N\sum x[n]^2$, is how audio software measures loudness (RMS).

Probabilities, expectations, energies, masses, light arriving at a pixel, the expected loss of a model: all are integrals. Most cannot be computed exactly, and a large part of computational science is about approximating them well — quadrature in low dimensions, Monte Carlo in high ones.