Product Measure and Fubini's Theorem · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Integrate one variable at a time

Mathematics · Calculus & Analysis · ages 22-24
Name ______________________   Date ____________

  1. Constant 2 on a rectangle 3 by 2. Double integral is constant times area. What is it?

    Answer: ______________

  2. What is the difference in hypotheses between Tonelli and Fubini?

    • Both need continuity
    • Both need boundedness
    • Tonelli needs nonnegativity; Fubini needs integrability
    • No difference

  3. Which statement gives the difference between the Tonelli and Fubini hypotheses?

    • Both theorems need continuity
    • Both theorems need boundedness
    • Tonelli needs nonnegativity while Fubini needs a finite absolute integral

  4. Exchanging the order of a double integral is licensed when the Tonelli or Fubini hypotheses hold.

    Circle one:   True   False

  5. Some sign changing function has two iterated integrals that both exist but differ, because its absolute integral diverges.

    Circle one:   True   False

  6. There exists a function where iterated integrals both exist but differ because the absolute integral diverges. Is this true?

    Circle one:   True   False

  7. Why can the order be exchanged for a continuous f on a closed rectangle?

    • Continuous on compact is bounded and integrable, so Fubini applies
    • Continuity alone never suffices
    • Rectangles are never measurable
    • Fubini never applies

  8. Why may you exchange the order for a continuous function on a closed rectangle?

    • A closed rectangle is compact, so the function is bounded and Fubini applies
    • Continuity alone always suffices with no further check
    • Rectangles are never product spaces

  9. The two iterated integrals of a function differ. Which hypothesis failed?

    • Continuity at a single point
    • Absolute integrability, since the integral of the absolute value is infinite
    • The domain being a rectangle

  10. Sum of coordinates on the unit square. Double integral is what?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Integrate one variable at a time W1-mt_1VzzVmMjTB-s1

  1. 12 · Area 3 times 2 equals 6, times constant 2 gives 12.
  2. Tonelli needs nonnegativity; Fubini needs integrability · Tonelli needs nonnegativity; Fubini needs integrability is the distinction, while continuity and boundedness are sufficient but not necessary.
  3. Tonelli needs nonnegativity while Fubini needs a finite absolute integral · Tonelli asks only for nonnegativity, while Fubini asks for a finite absolute integral.
  4. True · These two theorems are exactly what licenses the exchange of integration order.
  5. True · Sign changing examples show both iterates can exist yet disagree when absolute integrability fails.
  6. True · Classic sign changing counterexamples show this.
  7. Continuous on compact is bounded and integrable, so Fubini applies · Continuous on compact is bounded and integrable, so Fubini applies is correct, while the denials are false.
  8. A closed rectangle is compact, so the function is bounded and Fubini applies · Boundedness on the compact rectangle makes the absolute integral finite, so Fubini applies.
  9. Absolute integrability, since the integral of the absolute value is infinite · The exchange failed because the integral of the absolute value was infinite.
  10. 1 · Integral of x is 0.5 and of y is 0.5, summing to 1.
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