definite integral

The accumulated total of a function over an interval. Notation: `∫_a^b f(x) dx`. Geometrically, the _signed area_ between the graph of `f` and the x-axis from `a` to `b` (positive above, negative below). Operationally, the limit of a Riemann sum as the rectangle width shrinks to zero. Physically: the total displacement when `f(t)` is velocity, the total work when `f(x)` is force, the total charge when `f(t)` is current. The bounds matter: a definite integral is _one number_, not a function. Differentiation undoes it (via the fundamental theorem) by varying one bound and treating the result as a function of that bound.