I've often heard that the area under the curve is defined to be Riemann integral. Is it true? Or maybe we can prove that Riemann integral is equal to area under the graph of a function?
This book supports the first option. However, I think we might be able to prove that Riemann integral is equal to 'area' under the curve if we use axioms of area function.
You can define area in terms of measures, like Jordan measure, the value of Riemann integral on some interval equal to the Jordan measure of the area in the interval that bounded by the function and the x axis.