6.6 Properties of Definite Integrals

Core properties

∫[a to a] f dx = 0; swapping limits negates: ∫[a to b] f = −∫[b to a] f; and additivity: ∫[a to b] f + ∫[b to c] f = ∫[a to c] f.

Linearity & constants

Constant multiples factor out and integrals of sums split. Also ∫[a to b] k dx = k(b − a) for a constant k.

Key terms

Zero width

∫[a to a] f dx = 0.

Swap limits

∫[a to b] f = −∫[b to a] f.

Additivity

∫[a to b] + ∫[b to c] = ∫[a to c].

Constant integrand

∫[a to b] k dx = k(b − a).

Linearity

Constant multiples factor out; sums split.

Try an AP-style question

∫₀⁵ f(x) dx = 8 and ∫₀² f(x) dx = 3. What is ∫₂⁵ f(x) dx?

1. −5
2. 8/3
3. 11
4. 5