Im not exactly experienced with Calculus and math notation but I had question on Leibniz's notation:
If we have an equation $dy=2dx$, and we wanted to find $y$, we would take the integral of both sides:
$$y=\int{dy}=\int{2dx}=x^{2}+C.$$
From this we can see that taking the Integral $\int$ is the opposite of taking the derivative $d$ of a function, in this case $y$. In an extremely broad sense we could say that the integral $\int$ cancels out the $d$ in $dy$. Now here's where my question comes up:
If we wanted to find the area of a 2D shape like a square, in calculus notation it would look like this right?$$\text{Area}=A=\int{dA}$$
We are summing up the infinitesimal areas $dA$ inside the shape to find the whole area $A$. However, if we place the shape on a $(x,y)$ grid, we can also say:$$dA=dx\times dy.$$
The issue is, if we replace $dA$ in the equation we get $A = \int{dA} = \int{dx \times dy}$, which isn't right because to calculate $A$ we would have to integrate twice not once as the equation implies:
$$A=\int{dA}= \int \int (dx) dy.$$
But that implies that $d^{2}A=dx\times dy$ which doesn't seem right either conceptually.
I feel like I'm getting the notation wrong, or there's an identity or rule I'm missing. Maybe I made a mistake in my assumptions, I'm not sure. Any help would be appreciated.
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