1.8.3—Definite integrals

∫ₐᵇf(x)dx=F(b)−F(a) for an antiderivative F. It accumulates signed area: portions below the x-axis contribute negatively.

Substitute the upper and lower limits into the antiderivative, not the original integrand. Reverse limits change the sign, and equal limits give zero.

∫₀²(2x−1)dx=[x²−x]₀²=2; the graph has a small negative part but the net signed area is 2.

A definite integral is not always geometric area; use absolute values or split intervals when total area is requested.