Properties of Definite Integrals

Key idea

Linearity, additivity and reversal of limits.

In depth

Integrals can be split over intervals, ∫ₐᶜ = ∫ₐᵇ + ∫ᵇᶜ, reversing limits flips the sign, ∫ₐᵃ = 0, and constants factor out. For odd functions on [−a, a], the integral is 0.

Worked example

∫₋₂² x³ dx = 0 because x³ is odd.

Real-world uses

• Splitting a total into parts for piecewise cost or motion models

Common mistake

Forgetting the sign when swapping limits.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related advanced topics in Calculus

• Idea of a Limit – The value a function approaches as the input approaches a point.
• One-Sided Limits – Left and right limits must agree for the limit to exist.
• Limit Laws – Limits of sums, products and quotients are computed from the parts.
• Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
• Limits at Infinity – End behaviour of functions.
• Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.