I recently have to use derivates, antiderivatives, integrals more fluently and have to switch between them more and more.
I have a certain equation in front of me and I am a little confused about the connection between its forms. I wrote down a general rule for myself, which I would say is correct, but I couldn't verify it by research since it is not expliclitly mentioned like that:
Have I understand correctly that if:
$A=\int f(x) dx$
Then (this is where I do not doubt that this is true)
$\frac{dA}{dx}=f(x)$
And then:
$dA=f(x) dx$
And at this last one I am not confident about
(1)whether these 3 are always true together and
(2) how to correctly interpret it: "The function f(x) at a point x times an infinetly small change in x gives a square. This square's area is equal to an infinetly small change in A to that point" ?
I know there are dozens of questions about treating dy/dx as factors or not, but I haven't found an accurate answer to my question yet.
The Fundamental Theorem of Calculuslinks the relationship between differentiation and integration. Finding the definite integral of a function can be interpreted as the area under the graph of a function. It justifies procedure of evaluating an antiderivative at the upper and lower bounds of integration and taking the difference.
The first part of the theorem shows that indefinite integration can be reversed by differentiation. It also proves that for every continuous function, there is an antiderivative (integral).
Let $F$ be any antiderivative, or indefinite integral, for $f$ on $[a,b]$. If $f$ is a continuous function and is defined by $$F(x) = \int_x^a f(t)dt$$
Then $$\frac{d F}{dx} = f(x)$$
and therefore $$\frac{d}{dx}\int_x^a f(t)dt=f(x)$$
Now, when you come to 'solve' for an antiderivative using the separation method you described above, the easy answer to your question is that your definition for $dA/dx$ is correct; it means the derivative of $A$ with respect to $x$, and $dA$ and $dx$ are meaningless when written alone. It is worth ntoing that $dA$ is not the area of a saquare however, but the infinitesmal change in area of that square.
I should qualify the "not a meaningful expression." Something is only meaningless until somebody gives it a formal meaning. Then you hope that the meaning they gave it has useful properties (such as, that it relates to derivatives.
This has been done, and there is a good deal of mathematics that has gone into the theory of differentials, and it fits into integrals, and putting the differential $dx$ at the end of every integral also makes sense according to this theory, and so on.