Limit of integration can't be the same as variable of integration?

Solution 1:

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.

Solution 2:

It confuses a free variable and a bound variable. In effect you are saying "let $x$ range from $a$ to $x$ when taking the integral of $f(x)$".

It is also ambiguous. There is a risk some people might expect $\int_a^x f(x)dx = (x-a)f(x)$ in the same way as $\int_a^x f(x)dt = (x-a)f(x)$.

It is easier to show the problem as a sum. The sum of the first $n$ positive integers can be written $\sum_{i=1}^{i=n} i = \frac{n(n+1)}{2}$ but if you wrote it as $\sum_1^n n$, some people might expect the answer to be $n^2$. Meanwhile the following looks very strange $$1+2+3+\cdots+n+\cdots+(n-1)+n$$