# A question about the notation of integrals - does dx change if variable name is not x?

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?

• Yes, commonly you would change all occurrences of $x$. Sep 22 at 17:44
• 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 $\int f(a)\,dx = f(a)x + C$. Sep 22 at 17:45
• If you are integrating with respect to $a,$ the index would be $da.$ Sep 22 at 17:45
• $\int_A^B f(a)\,da=\int_A^B f(x)\,dx$. The symbols $a$ and $x$ are "dummy" variables. Sep 22 at 17:46

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