Limit of integration can't be the same as variable of integration? I am told that an expression like
$$
\int_a^x f(x)dx
$$
is not well formed, i.e. it should be
$$
\int_a^xf(t)dt
$$
or similar.

Why is it that the limits of integration can't depend on the variable of integration?
In mathematics, it's generally regarded as a bad idea for the same symbol to have two different meanings in the same expression. In this case, the variable being integrated with respect to effectively disappears, and a new variable (really two new variables, the bounds of integration) takes over. To call them the same thing can make things confusing sometimes (although not always). This is more of a stylistic than a strictly logical concern, at least in one variable.
your question is very valid and I see all the answers posted say a very comfortable no. I can give a physical situation where the limit is also a function of the variable in the integral (one which I am trying to solve myself). It goes like this:

The solar radiation intensity ($W/m^2$) at a point on the surface of the earth depends on the angle (theta) between the incident ray and the normal (radius vector) at the point. The exercise is to find the total radiant energy ($J/m^2$) over the period of a year. Now, theta as a function of time varies continuously and the limit of integration would cover the longitudinal sweep from sunrise to sunset. Bear in mind that this longitudinal sweep of daylight (from sunrise to sunset) varies with latitude of the point and also the time of year; thus the daily limit of integration is also a function of time (and this has to be summed up over a year).

I hope the problem statement is clear and I have worked out the angle relationships between theta, latitude, tilt of earth axis and angle swept by the earth in its orbit around the sun (taking the northern hemisphere summer solstice as the starting point).

It is a very practical problem and I'm sure that a solution to this exists that may have been discovered or is yet to be discovered.