Let $\phi:[0,1] \rightarrow \mathbb{R}^2$, with $\phi(t)=(x(t),y(t))$, a function satisfying the following assumptions:
(i) $x(t)$ and $y(t)$ are absolutely continuous;
(ii) $\phi(0)=\phi(1)$, the restriction of $\phi$ to $[0,1)$ is injective.
From Jordan curve's theorem we know that $R^2 \backslash \phi([0,1])$ is the union of two open connected sets, of each of one $\phi([0,1])$ is the boundary. Let $C$ be the closed bounded region "encircled" by $\phi([0,1])$. Groemer, Geometric Applications of Fourier Series and Spherical Harmonics, Section (4.1) states without proof that \begin{equation} m(C) = \int_{0}^{1} x(t) dy(t) , \end{equation}
where $m$ is Lebesgue measure in $\mathbb{R}^2$, and the integral on the right-hand side is intended a Riemann-Stieltjes integral.
Do you have some idea of the proof?
Some Notes. First of all, note that $\phi$ is a rectifiable curve, with length \begin{equation} L= \int_{0}^{1} \left| \phi'(t)\right| dt, \end{equation} where the integral is a Lebesgue integral (see e.g. http://www.math.binghamton.edu/loya/506-S12/506-1.pdf Theorem 8.31), and since rectifiable curves has zero Lebesgue measure (see e.g. Casper Goffman, "Rectifiable curves are of zero content", Mathematics Magazine 44 (1971), 179-180), then being $\phi([0,1])$ the topological boundary of $C$, we conclude that the indicator function $\mathbb{1}_{C}$ is Riemann integrable (by Vitali-Lebesgue Theorem, see e.g. Apostol, Mathematical Analysis) so that \begin{equation} m(C) = \int_{R^2} \mathbb{1}_{C}(x,y) dx dy, \end{equation} and the integral on the right-hand side is a Riemann integral (obviously the equality is still true if we intend this as a Lebesgue integral). Finally, note that, being $x(t)$ and $y(t)$ absolutely continuous, we have (see e.g. the post How to compute Riemann-Stieltjes / Lebesgue(-Stieltjes) integral? ) that \begin{equation} \int_{0}^{1} x(t) dy(t) = \int_{0}^{1} x(t) \dot{y}(t) dt, \end{equation} where the integral on the left-hand side is a Riemann-Stieljes integral, and the integral on the right-hand side is a Lebesgue integral. Note that if $x(t), y(t)$ are assumed to be smooth enough then the required equality follows from the version of Stokes' Theorem (called Green's theorem in dimension 2) for smooth manifolds with boundary. In this case, the differentiable version of Jordan-Brouwer Separation Theorem (for which see e.g. Guillemin and Pollack, Differential Topology) ensures that our set $C$ is a manifold with boundary, with boundary (in the sense of differential topology) $\phi([0,1])$. So the task is to prove here a version of Green's Theorem in the above setting, in which regularity assumptions are slackened.
The equality under discussion is used by Groemer to give a beautiful and elegant proof of the Isoperimetric Inequality, due essentially to Hurwitz, which uses Fourier series.
