I came across a question on a high school level exam paper. The question reads:

> 
> Show that$$
> \int\_a^b x^ndx=-\int\_b^a x^n dx.
> $$
> 
> 
> 

Of course, students are just expected to find the antiderivative and simply substitute the limits. However, this question drags me into deep thoughts: how do we formally define integrals where the upper limit is smaller than the lower limit? 

Clearly, we should have $\int\_a^b f(x)dx=-\int\_b^a f(x)dx$. But when I was trying to "prove" this from the definition of Riemann integrals, I find that if $a>b$, then dissections of the interval $[a,b]$ of the form $a=x\_0<x\_1<...<x\_n=b$ is impossible. **Apparently, we have *not* defined the integral $\int\_a^b f(x)dx$ for $a>b$ in real analysis books.** Lebesgue integrals don't help either, because it is integrating over **sets** rather than across two limits.

And here is the problem: for $a>b$, we have
$$
\int\_b^a f(x)dx=\int\_{[b,a]} f(x) d\mu,
$$
but $\int\_a^b f(x)dx$ is undefined.

Of course, we can say that, *by definition*, $\int\_a^b f(x)dx=-\int\_b^a f(x)dx$. But somehow, I think we should not brutally define it; we should make it look more natural, i.e., to prove it under some assumptions.