Definite Integrals & the Fundamental Theorem

Area as a limit of sums

Partition [a,b], form Riemann sums of rectangle areas, take mesh → 0. The definite integral ∫ₐᵇ f(x) dx is that limit when it exists. Signed area: below the x-axis counts negative.

Properties that save time

• ∫ₐᵇ f = −∫ᵇₐ f (flip limits flips sign).
• ∫ₐᵃ f = 0.
• Additivity over adjacent intervals: ∫ₐᶜ = ∫ₐᵇ + ∫ᵇᶜ.
• If f ≥ 0 on [a,b], the integral equals geometric area under the curve.

Worked example

∫₀² (3x² − 2x) dx = [x³ − x²]₀² = (8 − 4) − 0 = 4.