We define,

INT(a,b)f(x) dx = Limit of Riemann Integral (if it exists).

For function f defined on closed interval [a,b].

Note, under this strict definition, we require that f to be defined on [a,b] (and to be integratable). Hence a<b.

What about b>a?

Wedefine

INT(a,b)f(x) dx=-INT(b,a)f(x)dx.

Again, this IS NOT a theorem.

This is a definition.

And if a=b we define INT(a,b)f(x)dx=0.

For f defined at "a".