unit of integration vs unit of boundaries

Hi

Was just wondering:

In the expression

$\displaystyle \int_{a}^{b}f(x)dx$

it is assumed that a and b are values of x.

Yet when we do u-substitution, to get

$\displaystyle \int_{a}^{b}g(u)du$

we are still assuming that a and b are value of x (the boundaries of the area we are finding, as it were).

But really, shouldn't they also be converted to the corresponding values of u? The unit of integration is now u and everything else in the expression is now in terms of u, yet these boundaries are still referring to values of x. To change f(x) -> g(u) we needed some function mapping x to u, so we must also be able to map the boundaries to u as well.

The conventional method when using u-ubstitution to help with integration involves re-converting the final expression (e.g. 4u^2 +2u +c) back to x, and then substituting in the boundary values of x.

If the boundary values are converted from x-values to u-values, this reversal wouldn't be required before the final evaluation.

While I admit that it is probably just easier to do it the conventional way, my question is whether or not the procedure outlined above is actually correct.

Cheers