I think that using the arc length differential $ds$ in a line integral finds the area like in https://en.wikipedia.org/wiki/File:Line_integral_of_scalar_field.gif, but I cannot imagine what the area using $dx$ or $dy$ is.
Sometimes the question asks "Evaluate
a) $$\int_C xy^2\,dx$$ b) $$\int_C xy^2\,dy$$ c) $$\int_C xy^2\,ds$$
where the path $C$ is defined by $x=4\cos(t)$, $y=4\sin(t)$, $0\leq t\leq \pi/2$."
I know how to evaluate these, but what do they mean geometrically? Specifically, a) and b) since their differentials are $dx$ and $dy$.