When can we consider area between functions? When can we consider area between functions? Do they always need to be continuous or do we just define it as of the form $$\int_{[a,b]}|f-g|$$ for any function $f$ and $g$ if it exists?
I am curious about the definition of the area, especially area between two functions in calculus?
 A: We can define it in the way you suggest. The functions can be wildly discontinuous and still allow it to be defined. For example, take any real-valued function $f$ defined on $[a,b]$ and define $g(x):=f(x)-1$ for all $x,$ so that the area between $f$ and $g$ on $[a,b]$ is $b-a.$
Edit: The argument can be made that this notion of area isn't meaningful if $f$ and $g$ aren't integrable over $[a,b],$ but we can certainly still define $\int_{[a,b]}|f-g|$ in many more circumstances than that.
A: Limiting the discussion just to finite functions over finite intervals (for simplicity), the integral of a function can be interpreted as the algebraic area between the graph of the function and the $X$-axis. For that to be meaningful the function must be integrable. Now, classically one speaks of Riemann integrability, which is quite a large class of functions, including all of the continuous ones but many many more. In fact a function is Riemann integrable if, and only if, the set of its discontinuities is 'small'. The technical term is being of Lebesgue measure zero. More generally, the class of Lebesgue integrable functions includes all the Riemann integrable ones and many many more. There isn't a simple criteria though for Lebesgue integrability. 
A: Perhaps this question should be construed as meaning this:
Let $A = \{ (x,y)\in\mathbb R^2 : a\le x\le b \text{ and $y$ is between $f(x)$ and $g(x)$} \}.$
Then $\displaystyle (\text{2-dimensional Lebesgue measure of } A) = \int\limits_{[a,b]} |f-g| \text{ ?}$
Measurability of $|f-g|$ is weaker than measurability of $f$ and of $g,$ and is enough for the existence of the integral on the right.
If $f$ and $g$ are non-measurable, does $A$ necessarily fail to be measurable? I'm not sure right now, but certainly if $f$ and $g$ are measurable then $A$ is measurable. In that case,
\begin{align}
\iint\limits_{[a,b] \, \times \, \mathbb R} 1_A(x,y)\, d(x,y) & = \int\limits_{[a,b]} \left( \int\limits_{\mathbb R} 1_A(x,y) \, dy \right) \, dx \text{ by Fubini's theorem} \\[12pt]
& = \int\limits_{[a,b]} |f(x)-g(x)| \, dx.
\end{align}
