Integrate one variable at a time

Mathematics · Calculus & Analysis · ages 22-24

Name ______________________ Date ____________

Integrate f(x, y) = 2x over 0 to 1 in x and 0 to 4 in y.
Answer: ______________
For f(x, y) = x minus y on the unit square, which theorem can license swapping the order?
- Fubini, since the absolute integral is finite
- Tonelli, since f is measurable on the square
- Tonelli, since the square is a closed rectangle
- Neither, since f takes negative values
Which statement gives the difference between the Tonelli and Fubini hypotheses?
- Tonelli needs a finite absolute integral while Fubini needs nonnegativity
- Both theorems need the function to be continuous and bounded
- Tonelli needs nonnegativity while Fubini needs a finite absolute integral
Exchanging the order of a double integral is licensed when the Tonelli or Fubini hypotheses hold.
Circle one: True False
Some sign changing function has two iterated integrals that both exist but differ, because its absolute integral diverges.
Circle one: True False
For f(x, y) = (x squared minus y squared) divided by (x squared plus y squared) squared on (0, 1] by (0, 1], the two iterated integrals are equal. Is this true?
Circle one: True False
On the unit square, f(x, y) = y times e to the (x y). Integrating in x first gives e minus 2. What does integrating in y first give, and why?
- 2 minus e, because swapping the order flips the sign
- A different value, since integrating in y first has no closed form
- e minus 2, because f is continuous and bounded on the square
- No value, since the iterated integrals of e to the xy diverge
Why may you exchange the order for a continuous function on a closed rectangle?
- A closed and bounded rectangle makes the function bounded, so Fubini applies
- Continuity alone licenses the swap on every domain, including unbounded ones
- A closed rectangle is not a product of intervals, so Tonelli is needed instead
A sign changing function on a square has two iterated integrals that both exist but 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
f(x, y) = x times e to the minus y, with x from 0 to 2 and y from 0 to infinity. Tonelli lets you integrate in either order. What is the value?
Answer: ______________

Answer key

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

Integrate one variable at a time W1-mt_1VzzVmMjTB-s1

4 · Integrating in y first gives 2x times 4, which is 8x. Integrating 8x from 0 to 1 gives 4.
Fubini, since the absolute integral is finite · f changes sign, so Tonelli does not apply. But |x minus y| is at most 1 on the unit square, so the absolute integral is finite and Fubini applies.
Tonelli needs nonnegativity while Fubini needs a finite absolute integral · Tonelli asks only for nonnegativity, while Fubini asks for a finite absolute integral.
True · These two theorems are exactly what licenses the exchange of integration order.
True · Sign changing examples show both iterates can exist yet disagree when absolute integrability fails.
False · Integrating in y first gives pi over 4, and in x first gives minus pi over 4. The integral of |f| is infinite, so Fubini does not force the orders to agree.
e minus 2, because f is continuous and bounded on the square · f is continuous on the closed unit square, so it is bounded and its absolute integral is finite. Fubini then says both orders agree, so the y-first order also gives e minus 2.
A closed and bounded rectangle makes the function bounded, so Fubini applies · Boundedness on the closed and bounded rectangle makes the absolute integral finite, so Fubini applies.
Absolute integrability, since the integral of the absolute value is infinite · Tonelli was never available because the function changes sign. If the integral of |f| were finite, Fubini would make both orders agree, so that integral must be infinite.
2 · f is non-negative, so Tonelli applies. Integrating in y first gives x times 1, and integrating x from 0 to 2 gives 2.