The first rigorous integration theory in due to Eudoxus and Archimedes. It is called the method of exhaustion, and it allowed Archimedes to find the volumes of the balls, pyramids, cones, areas of segments of parabolas etc.
(By the way, it was proved in 20th century that one cannot find the volume
of a pyramid by pure geometric methods, some form of integral is indeed necessary here).
In the modern era, the first rigorous definition of the definite integral is due to Cauchy.
Cauchy's definition of the integral is close to that of Riemann, the main difference is that
he takes equidistant points in the partition to define the integral sums, and assigns
specific points where the values of the function are taken. This gives the same result for
continuous functions as Riemann's integral.
Cauchy's approach is used sometimes in elementary calculus courses, because it is conceptually simpler than the Riemann approach.
It is not true that Riemann integral is "not as useful" as Lebesgue's integral. Because in many cases the functions we integrate are continuous. For example in all books on Complex Analysis, Riemann integral is used. It is also taught in almost all courses of real analysis, with very few exceptions, before the Lebesgue's integral, even in the Bourbaki course.
Sources. Medvedev, Development of the concept of integral, Moscow 1974.
I. N. Pesin, Development of the concept of integral, Moscow 1966 (There is an English translation).
N. Bourbaki, Elements of the history of mathematics. Translated from the 1984 French original by John Meldrum. Springer-Verlag, Berlin, 1994.
N. Bourbaki, Functions of a real variable, Elementary theory. Translated from the French, Springer 2004.