When doing a surface integral like "$\phi=k\cdot\int \int _ {\text{lateral surface}} \frac{\sigma}{r} da$" along the lateral surface of a cone it is necessary to find $da$ in terms of some iterable value. I know in this case I can find $da = dl \cdot R \cdot \frac{l}{L}\cdot d\theta$ from geometry.
My question is if this alternate approach works:
Given a cone with slant height $L$ and base radius $R$, we can set up the following integral to solve for surface area:
$$\text{surface area} =\int\int_{\text{lateral surface of cone}}1 \cdot da=\pi R L$$
Now how do I solve for $da$?
The answer is no, that does not work. The result of an integration depends on the global behavior of the integrand. That is, the value of $$\iint_S f\,dA$$ depends on the behavior of $f$ at every point on $S$, and more particularly for your question, on the shape of $S$ around each of its points. The result is a single number, so it contains much less information than is available from the full shape of $S$.
But $dA$ is local in its nature. the "value" of dA at a point in $S$ depends only on how the surface is shaped near that point. The very data needed to determine this is lost when you integrate over the entire surface. So you cannot determine the local $dA$ from the global $\iint_S dA$.
In special cases such as the cone where the surface has the same shape almost everywhere, then it is possible to determine $dA$ from the area formula, but this amounts to finding $dA$ from the geometry, just as you already did.