A definite integral is defined as a limit of Riemann sums:
$$\int f \;dx:= \lim_{\|P\| \to 0} \Sigma f(c_i)\Delta x_i$$
Would the following be equal to $0$ in all cases?
$$ \int f \;d^2x:= \lim_{\|P\| \to 0} \Sigma f(c_i)(\Delta x_i)^2 = 0?$$
My guess is yes but I don't know how you would rigorously prove this. Consider a parameterization of a curve in 2-space, $c(t):= (x(t), y(t))$. The line integral of a vector-valued function $\vec{f}$ over this curve would be
$$ \int \vec{f}\cdot \; d\vec{s}$$
where $d\vec{s}$ represents a line element. $d\vec{s}$ is taken as $\vec{c}\;'(t) dt$. Wouldn't a better approximation go to the second-order term of the taylor series of $c$? So $d\vec{s} = \vec{c} \;' dt + \vec{c}\; '' (dt)^2$. This makes me think that definite integrals with higher order differentials are $0$? Is this reasoning correct? (likewise with change of variables - we only need the Jacobian because the "principal part" of the "change" is what is important. All higher-order terms under an integral sign would tend to $0$?)