Is this proof of integration as anti-derivative right?

Question

I recently learned calculus and was thinking about why Integral is taken as anti-derivative. I came up with the following proof. I want to know if this is right or not and also if there is a better method for this.

Let $F(x)$ be a curve and its derivative be $f(x)$ i.e. $\frac{dF(x)}{dx} = f(x)$ . Let us divide our curve $f(x)$ to many little sections each of width $dx$ where $\lim \limits_{dx\to 0}$. Each of these tiny sections have an area of $dA$.
Now,
$dA$ = $f(x)dx$ [Where f(x) is the height of curve at any point where x-coordinate = x]
=> $\frac{dA}{dx}$ = $f(x)$
But,
$f(x)$ = $\frac{dF(x)}{dx}$

So, $dA = dF(x)$
Hence,
$$\lim \limits_{dx\to0} \sum_{x=a}^b f(x)dx = \sum_{x=a}^b dF(x) $$
$$\int_{x=a}^{x=b} {f(x)dx} = \sum_{x=a}^b dF(x) $$

Finally, as $dF(x)$ which is tiny change in $F(x)$ = $dA$ which is a tiny area in our curve, total area under our curve, say from $x=a$ to $x=b$ must be total change in $F(x)$ from $x=a$ to $x=b$. Hence, $$\int_{x=a}^{x=b} {f(x)dx} = [F(b) - F(a)]$$
NOTE : Since, I haven't got much experience in proof-writing, there may be some lack of interpretation and explanation above which I am happy to correct