superposition integral
A complicated load is just a crowd of tiny point loads standing shoulder to shoulder. If you know what one point load does — the point-source response — then the full response is the sum of all those little responses, each scaled by how strong its point is. When the points form a continuum, that sum becomes an integral. The superposition integral is that integral: the formula that assembles the answer for any source out of the Green's function.
Written out, it says u(x) = integral of G(x, s) f(s) ds over the domain. Read it as a weighted accounting: at every source location s the load has density f(s); each such bit contributes f(s) ds worth of point source, which the system answers with G(x, s); add up the contributions over all s and you get the field u at the observation point x. The legality of this step is the superposition principle — it holds precisely because L is linear, so L u = integral of L G(x, s) f(s) ds = integral of delta(x - s) f(s) ds = f(x). The delta sifts out f(x), and the equation is satisfied.
This is the payoff of the whole Green's-function program: it converts an entire family of boundary-value problems (one for each source f) into a single integration against a fixed kernel. In a translation-invariant, whole-space setting the superposition integral is literally a convolution, u = G * f, which is why Fourier and Laplace transforms — which turn convolution into multiplication — pair so naturally with Green's functions in signals, optics, and field theory.
With G(x, s) = x(1 - s) for x < s and s(1 - x) for x > s on [0, 1], take f(s) = 1. Then u(x) = integral from 0 to x of s(1 - x) ds + integral from x to 1 of x(1 - s) ds = x(1 - x)/2 — recovering the uniformly loaded string by superposing point responses.
Splitting the integral at x reflects that G changes formula as the source crosses the observation point.
Superposition is exclusively a property of linear problems. For a nonlinear equation you cannot add the responses to separate sources, so there is no Green's function and no superposition integral — this is the single sharpest limitation of the whole method.