When taking the double integral of a function in terms of the same variable $x$, should I write $\int\int y~\text d^2x$ or $\int\int y~\text dx^2$, and why?
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Neither is correct. You can't integrate wrt a squared differential - it's a bit of abuse of notation.
See the discussion here: Is there a better notation for integrating twice?
For indefinite integrals you'll need to write:
$$\int\int y \;dxdx = \int F_y(x) + C\; dx = F_{F_y}(x) + Cx + D$$
Where I'm just using $F_y$ to denote "The antiderivative of y" and making the dependence on $x$ explicit.
Ross Millikan give a good explanation here: https://math.stackexchange.com/a/2084938/632875
For definite integrals, you want to use different dummy variables for each integral to make it clear what you are integrating over each time:
$$\int_0^x \int_0^t y(z)dzdt = \int_0^x[F_y(t) -F_y(0)]dt = F_{F_y}(x)-F_{F_y}(0) - xF_y(0)$$