THE PROBLEM
I am dealing with the following object:
$$I = \displaystyle\int_C f(x,y)\,dxdy,$$
where $C$ is a curve on the Cartesian plane and $f: \mathbb{R}^2\rightarrow \mathbb R$ is any Riemann integrable function in a neighbourhood of $C$. It seems to me that this object must be interpreted as the integral over the curve $C$ of a rank two, symmetric linear form. There is no other way to interpret this object: it is not a line integral of $f(x,y)$, nor a two variable integral over a two-dimensional domain of the plane. If what I am saying makes sense (hope someone could confirm or refute what I have previously said), I would expect that the object $I$ should be a 1-form.
Now, suppose the curve $C$ could be parametrized by some parameter $\theta$, i.e.,
$$C:\;\;\;\gamma(\theta) = (x(\theta),y(\theta)),\hskip1cm \theta\in[a,b]\subset\mathbb R,$$
Am I right if I write $I$ as follows?
$$I = \displaystyle\int_a^b f(x(\theta),y(\theta))\,x'\,y'\,d\theta\,d\theta,$$
where $x'$ and $y'$ are the first derivatives of the components of $\gamma$. I know, it seems a little bit odd but I think it is natural in the context of symmetric rank two tensors where an object like $d\theta\,d\theta$ is written via the tensor product as $d\theta\otimes d\theta$.
Now, if fortunately what I have said makes sense and is correct, how can I compute the integration if I know the function $f(x,y)$? In other words, what theorem or concept I have to invoke in order to perform the integral
$$\displaystyle\int_a^b f(x(\theta),y(\theta))\,x'\,y'\,d\theta\otimes d\theta,$$
and obtain as a result the explicit 1-form I have supposed at the beginning?
Thanks a lot!
EDIT 1: Answering Ted Shifrin
The precise problem can be found on page 129 of "The Method of Moments in Electromagnetics" of W.C.Gibson (third edition). I write below the equation which made me formulate the question in this post. This is
$${\omega\mu \over 4}\int_C {\bf J}({\bf\rho}') H_0^{(2)}(k|{\bf\rho}-{\bf \rho}|)d{\bf \rho}'={\bf E}^{i}({\bf \rho}).$$
In this equation, $\omega, \mu$ and $k$ are electrodynamical parameters, $H^{(2)}_0$ is an Hankel function of the second kind, $\bf J$ and $\bf E$ are the current density and the incident electric field respectively (they are vectors, but each component can be treated separately). The integration variable is $\rho' = (x',y')$ as one can find at page 137 of the above text book in Eq. (6.77) (hence a vector of the Cartesian plane). Finally, the curve $C$ is a "contour of arbitrary shape" paraphrasing Gibson. Everything is made clear by just looking at the FIGURE 6.6 at page 138 of the book in question: the contour is a curve of ${\mathbb R}^2$ while the integration variable $\rho'$ is a vector of the same plane.
How would you interpret the equation above?