The definite integral of f(x) from a to b geometrically represents the signed area between y = f(x) and the x-axis. When f(x) >= 0, the integral equals the area directly. When f(x) < 0, the integral gives a negative value, so the geometric area is the absolute value. This distinction is crucial: the integral from 0 to 2pi of sin(x) dx = 0 (positive and negative areas cancel), but the total area between y = sin(x) and the x-axis from 0 to 2pi is 4 (two arches, each with area 2). Always ask yourself: does the problem want the signed integral or the geometric area? For geometric area, split the integral at every zero of f(x) in [a, b] and sum the absolute values.