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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?

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    It's a bit like writing $\sum_{i = 1}^i f(i)$. What is it supposed to mean?2012-02-14
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    There is no real reason why it shouldn't (in the end the integration variable is a dummy variable) -- however, choosing the integration variable to be the same as one of the limits opens many doors for error...2012-02-14
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    Try $\frac{d}{dx}$ on the first one. No confusion?2012-02-14
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    You *can* do as you have in the first integral, but you *shouldn't*. The x's play entirely different roles when you use x to denote both a limit and a [dummy variable](http://en.wikipedia.org/wiki/Integral#Terminology_and_notation)2012-02-14
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    Poor style, but in principle OK.2012-02-14
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    In the world of programming, this is a source of confusion for much the same reason. In fact, it can be used to intentionally make the code extremely difficult to understand--a process known as "obfuscation". The strict rules of scope etc make such code technically unambiguous, but it is often very difficult for humans to read.2014-06-30

2 Answers 2

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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 $$1+2+3+\cdots+n+\cdots+(n-1)+n$$

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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.