According to my understanding, the formula to calculate the signed area between two functions (or axes, as when finding "area under a curve") $f(x)$ and $g(x)$ from $a$ to $e$ is $\int_a^e f(x) - g(x)\ dx$, while the formula to calculate the unsigned version of the same area is $|\int_a^b f(x) - g(x)\ dx| + |\int_b^c f(x) - g(x)\ dx| + |\int_c^d f(x) - g(x)\ dx| + |\int_d^e f(x) - g(x)\ dx|$, where $b$, $c$, and $d$, are values of $x$ between $a$ and $e$ where the functions intersect.
Is it possible to extend these formulas to work when $f(x)$ and $g(x)$ are relations, but not necessarily functions? We can assume they're well-behaved enough to not interfere with the ability to perform the appropriate integrals. 
In the simplified example image shown above, the integrand from $a$ to $b$ should be $f(x) - g(x)$ (or $g(x) - f(x)$ since each integral is inside an absolute value function). However, $f(x)$ is a multifunction, and from $b$ to $c$, the correct integrand is the difference between two outputs of $f(x)$ (in fact, it must be a particular pair of outputs, which would come into play for example if the above $f(x)$ is some sort of inverse periodic multifunction like arccos).
Clearly this is obvious when the graph is drawn, but is there a way to modify the formulas to work without graphing every problem?