Fubini’s Theorem
Iterated integrals equal double integrals.
Key idea
Iterated integrals equal double integrals.
In depth
If f is integrable on a product space, the double integral equals either iterated integral: ∫∫f dx dy = ∫(∫f dy)dx. Integrability is needed; for non-negative f (Tonelli) no condition is needed.
Worked example
∫₀¹∫₀¹ xy dx dy = 1/4 in either order.
Real-world uses
- Swapping the order of integration in double integrals when computing volumes and expectations
Common mistake
Swapping the order without absolute integrability, which can give different answers.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Fubini’s Theorem?
Iterated integrals equal double integrals. It is a topic at the expert level in Analysis & Topology.
Where is Fubini’s Theorem used in real life?
Real-world examples: Swapping the order of integration in double integrals when computing volumes and expectations.
What should I learn before Fubini’s Theorem?
Make sure you are comfortable with Fatou’s Lemma, the previous topic in Analysis & Topology.
What is a common mistake with Fubini’s Theorem?
Swapping the order without absolute integrability, which can give different answers.
How can I practise Fubini’s Theorem?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Fubini’s Theorem?
Move on to Lp Spaces next.
Related expert topics in Analysis & Topology
- Axioms of the Real Numbers – ℝ is a complete ordered field.
- Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
- Archimedean Property – For any real x there is an integer n > x.
- Density of Rationals – Between any two reals there is a rational.
- Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
- Limit Theorems for Sequences – Algebra of limits and squeeze theorem.