True or False question for Fubini's Theorem

Question

a) Is it true that $$\iint_{R} f(x,y)dydx=\iint_{R} f(x,y)dxdy$$

I thought this would be true because of Fubini's Theorem however Fubini's Theorem requires $f(x,y)$ to be continuous on $R$. There is no such condition in this problem. Therefore it is false?

b) If $R$ is the rectangle $0\leq x\leq a, 0\leq y\leq b$ and $S$ is the rectangle $-a\leq x\leq 0, -b \leq y\leq 0$ then

$$\iint_{R} f(x,y)dA=-\iint_{S} f(x,y) dA$$

I think this is false and the counter example I would use is

$f(x,y)=1$

Then $\int_{0}^{b}\int_{0}^{a} dydx = \int_{-b}^{0}\int_{-a}^{0}dydx$

Is this correct?

There is no continuity in Fubini's Theorem. Where did you read Fubini's Theorem from? — Kavi Rama Murthy, May 24 '22 at 23:17
https://tutorial.math.lamar.edu/classes/calciii/IteratedIntegrals.aspx — user130306, May 24 '22 at 23:20
Well, the result is true without continuity but I can prove it only using Measure Theory. — Kavi Rama Murthy, May 24 '22 at 23:22
hmmm but I am in a vector calculus course, I have not learned any measure theory. So is the answer true for me? — user130306, May 24 '22 at 23:24
In the link that I posted, it says that $f(x,y)$ needs to be continuous, but that is only over a rectangle. Is that why? — user130306, May 24 '22 at 23:25

score 1 · Accepted Answer · answered May 30 '22 at 05:45

a) Is it true that $$\iint_{R} f(x,y)dydx=\iint_{R} f(x,y)dxdy$$

I thought this would be true because of Fubini's Theorem however Fubini's Theorem requires $f(x,y)$ to be continuous on $R$.

The continuity version of Fubini's Theorem in $\mathbb R^2$ is a special case.

The above equality does hold as long as $\int f(x,y)\,\mathrm dx$ and $\int f(x,y)\,\mathrm dy$ both exist on rectangle $R$'s respective edges.

b) If $R$ is the rectangle $0\leq x\leq a, 0\leq y\leq b$ and $S$ is the rectangle $-a\leq x\leq 0, -b \leq y\leq 0$ then

$$\iint_{R} f(x,y)dA=-\iint_{S} f(x,y) dA$$

I think this is false and the counter example I would use is

$\int_{0}^{b}\int_{0}^{a} dydx = \int_{-b}^{0}\int_{-a}^{0}dydx$

Is this correct?

Yes (ignoring your typo where you interchanged $x$ and $y).$

True or False question for Fubini's Theorem

1 Answers1