We know that Riemann integral is defined to be the area between the graph of a function $y=f(x)$ and $x$-axis. Actually, many people state the definition works in both ways (area under the curve is defined to be equal to Riemann integral).
If a Riemann integral of $f$ exists, then Lebesgue integral also exists and is equal to Riemann integral. However, there are Lebesgue-integrable functions that are not Riemann-integrable.
Does Lebesgue integral also give the area under the curve of such functions? Can it be proven formally? Actually this is an important question, because it depends on whether the area under a curve is defined as the value of integral or not.