When we say "$f$ is Riemann Integrable on $[a,b]$", do we assume that $a \leq b$?

Question

Suppose $a < b$ and $f$ is Riemann Integrable on $[a,b]$. Then by definition, $\int_b^a f = -\int_a^b f$. Do we then say that "$f$ is Riemann Integrable on $[b,a] = \emptyset$" ?

Apologies for this silly/pedantic question, I was just interested to know.

Answer 1

No, both the expression $$\int_a^b f(x)dx$$ and the expression $$\int_b^a f(x)dx$$ refer to a function $f$ that is integrable on the interval $[a,b]$, or, if you really want to be pedantic, then you could say the interval $[\min\{a,b\}, \max\{a,b\}]$.

The only thing is that you have two cases:

1. If $a<b$, then $\int_a^b f(x)dx$ is defined as the limit of Riemann sums while $\int_b^a f(x)dx$ is defined as $$\int_b^a f(x)dx=-\int_a^b f(x)dx$$
2. If $a>b$, then it all switches places, $\int_a^b f(x)dx$ is now defined as $$\int_a^b f(x)dx=-\int_b^a f(x)dx,$$ while $\int_b^a f(x)dx$ is defined as the limit of Riemann sums.