TITLE: What is $dx$ in integration?
QUESTION [177 upvotes]: When I was at school and learning integration in maths class at A Level my teacher wrote things like this on the board.
$$\int f(x)\, dx$$
When he came to explain the meaning of the $dx$, he told us "think of it as a full stop".  For whatever reason I did not raise my hand and question him about it.  But I have always shaken my head at such a poor explanation for putting a $dx$ at the end of integration equations such as these.  To this day I do not know the purpose of the $dx$.  Can someone explain this to me without resorting to grammatical metaphors?

REPLY [2 votes]: It's very bad teaching to say that students should see the $dx $ part of an integral as something like a full-stop. Having met some poor math teachers myself (and some excellent ones, thankfully) I can empathize. Can I suggest that you just 'park' your old teacher's suggestion and look afresh - from a new perspective - at calculus ?
Before taking calculus, your should have been taught about the limits of functions. Otherwise differentiation and integration would be impossible to understand in the context of algebra and geometry. I strongly recommend Teach Yourself Calculus by P. Abbott. Old editions from 1960s are available on Amazon secondhand. (Please do NOT get the 'new' version by a chap called O'Neill - this is just a revenue hijack, as far as I can see.) 
After getting a good handle on limits of functions, you'll soon see how 
         $$ \lim_{\Delta x \to 0}{\frac{\Delta y}{\Delta x}} = \frac{dy}{dx}$$
In physical terms, $dx$ might be seen as the size of the biggest (but of course still quite miniscule in real-life terms) change to a variable, $x$, that will provide us with a means to make a precise estimation of the resulting change, $dy$, to a function $y = f(x)$. We can make this estimation using the differential of the function, $y = f(x)$, with respect to variable $x$. 
$$ \frac{dy}{dx} = \frac{d[f(x)]}{dx}  = f'(x) $$
Hence 
$$  dy = (\frac{dy}{dx}) \ dx = f'(x) \ dx $$
Using algebra, we can work out what $ f'(x) $ or $ \frac{dy}{dx}$ is for any function via:
$$ f'(x) = \frac{dy}{dx} = \lim_{\Delta x \to 0}{\frac{f(x + \Delta x) - f(x)}{\Delta x}} $$
After doing a few of these one can set up a set of 'rules' by which different types of functions (e.g. polynomials, trigonometrics, product of functions, quotient of functions, etc) can have their differential function written out almost at sight. So if we are given a function $f(x)$ we can quickly write out its differential, $f'(x)$.
Differentials like $\frac{dy}{dx}$ tell us about the rate of change of a function compared to that of its dependent variable. But we sometimes need to do the opposite, i.e. find the function whose differential function, $f'(x)$, we happen to know.
For example, we may need to find an overall change to that function as its dependent variable changes over a continuous range of values between $x_1$ and $x_2$. To do this we use the previous relation
$$ dy = d[f(x)] = f'(x) \ dx $$
This equation tells us that each part of the overall change in $y = f(x)$ as $x$ goes from $x_1$ to $x_2$ is the product of the rate of change of y with x evaluated at a point in that range multiplied by the differential in $x$, $dx$. Since we do not know the absolute value of $dx$ we cannot do it precisely by calculations of each $dy$. But we can use our 'rules' for differentiating a function in reverse to find the original function $f(x)$ whose $\frac{dy}{dx}$ equates to the function $f'(x)$. Applying $x_1$ and $x_2$ as arguments in this function will then provide us the overall change in $y$ as $x$ changes from $x_1$ to $x_2$, i.e.
$$  y = f(x) = \int{f'(x) \ dx}$$
where the $\int$ symbol signifies the reverse process (called integration) to the differentiation of a function.
Inserting $x_1$ and $x_2$ into the final function and subtracting gives the overall change in the function $y$ as $x$ changes from $x_1$ to $x_2$, i.e.
$$  \Delta y = f(x_2) - f(x_1) $$
So if your old teacher said that $dx$ in an integral should be seen as a full-stop, he was not correct. The $dx$ component must be there for the integral to have meaning since every integral is a product of a differential function and a differential in the dependent variable.
A thought has just occurred to me: perhaps your old teacher meant that there should be a full-stop between the $f(x)$ and the $dx$ ? That would make sense, as in algebra a dot (full-stop or period) is the operator for a multiplication - which is what $f(x) \ dx$ is when inside an integral . . .
$$  y =  \int f'(x) \ . dx $$
I was going to suggest a thorough caning for that old teacher but now maybe the old devil wasn't so wrong after all . . . though this "insight" alone hardly takes us very far in understanding integrals.