A definite integral has limits — and gives a number, not a function.
Drag the right edge
y = x·xarea [0, 1.5] ≈ 1.13
From function to number
An *indefinite* integral gives a family of functions; a *definite* integral pins down the limits a and b and hands back a single number — the net area between the curve and the x-axis over [a, b]. Drag the edge above and watch the area readout grow as b moves right.
Area below the x-axis counts as negative. ∫₀^(2π) sin x dx = 0 because the bump above cancels the dip below.
Properties worth memorising
- ∫ₐᵃ f dx = 0 — zero width, zero area.
- ∫ₐᵇ f dx = −∫ᵇₐ f dx — swapping limits flips the sign.
- ∫ₐᵇ f dx + ∫ᵇᶜ f dx = ∫ₐᶜ f dx — areas add over adjacent intervals.
- Constants slide out, sums split apart — same as indefinite integrals.
Your turn
∫₁³ 2x dx = ?
Fundamental theorem
Try it
∫₀² x² dx
[x³/3] from 0 to 2 = 8/3 − 0 = 8/3.