My sketchy understanding of the (no doubt long) history of integration theory is that the first integration theory was created by Riemann as part of his work on trigonometric series ("Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe", according to Wikipedia).
Other integration theories followed. The standard one today is the Lebesgue integral, which is used, among other places, in the theory of probability.
The Riemann integral is not as useful as the Lebesgue integral, but is still a "modern" integration theory, and is comparable to the Lebesgue integral.
My question is whether this was really the first "modern" integration theory. Did Riemann come up with it out of the blue? Were there precursors? And if it was his original creation, how much is known about how he developed the notion?