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I have a question about the notation of integrals, I apologize if it is stupid. Usually the notation of integrals goes like this:

$$ \ \int f(x) dx \ $$

But let's say that for whatever reason we have chosen to name a variable something other than $x$ - for example, $a$. Would then the notation change to this:

$$ \ \int f(a) da \ $$

Or would the $dx$ term remain unchanged:

$$ \ \int f(a) dx \ $$

or some third option?

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    $\begingroup$ Yes, commonly you would change all occurrences of $x$. $\endgroup$
    – user85667
    Sep 22 at 17:44
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    $\begingroup$ Just as with integrals, the "dx" tells you what is the variable you are integrating with respect to. $\int f(x)\,dx$ will give you the same answer as $\int f(a)\,da$, but using $x$ as the name of the variable in the first, and $a$ as the name of the variable in the second. $\int f(a)\,dx$ would be integrating a function whose value does not depend on $x$ with respect to $x$, so $\int f(a)\,dx = f(a)x + C$. $\endgroup$ Sep 22 at 17:45
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    $\begingroup$ If you are integrating with respect to $a,$ the index would be $da.$ $\endgroup$
    – Doug M
    Sep 22 at 17:45
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    $\begingroup$ $\int_A^B f(a)\,da=\int_A^B f(x)\,dx$. The symbols $a$ and $x$ are "dummy" variables. $\endgroup$
    – Mark Viola
    Sep 22 at 17:46

1 Answer 1

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Remember that we usually get into integrals via Riemann sums, i.e. by taking

$\int f(x)\ dx = \lim \sum_n f(x_n) \Delta x_n$

with appropriate choices of partitions and limits. So the notation $\int f(x)\ dx$ can be thought of as "the sum of lots of bits of $f$ evaluated at points $x$, multiplied by small intervals of size $dx$".

So the answer is that $x$ is an arbitrary choice of variable, but it has to be kept consistent across the integral - if we measure $f$ at values of $y$, then we need to look at intervals of $dy$ as well, so $\int f(x)\ dx = \int f(y)\ dy$.

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