Why change the sign of the integral when switching the limits of integration?

Solution 1:

$\int_a^b f$ is a misleading notation. What is really defined is the integral over the interval, the set. There is no "orientation" involved. Think of this as $\int_{[a,b]} f$ instead.

Now, we define the symbols $\int_a^b f:=\int_{[a,b]} f$ and $\int_b^af:=- \int_{[a,b]} f$. Why? Because this is convenient. This makes a lot of formulas easier to write, and a lot of cases can be condensed in a single statement (for instance, as commented, the change of variables formula).

Solution 2:

One thing that's definitely simpler with the standard notation is this: The formula $$\int_a^bf(x)dx+\int_b^cf(x)dx=\int_a^cf(x)dx$$is valid with no restrictions on $a,b,c$. With your revision it would be true only if $a\le b\le c$ or $c\le b\le a$. I've always assumed that this was the reason for the convention being the way it is.