Integration and Euclidean geometry

Question

What intrinsic relationship is there between (riemann) integration and Euclidean geometry that enables one to get area under the curve as a integral? This can only be related to the definition of riemann integral as a limit of sums of areas of rectangles; or is it something else.

Answer 1

The Riemann integral is defined in such a way (through a limit of sums of areas of rectangles, as you wrote) that the area under the curve of a non-negative function is equal to the integral of that function.

Answer 2

A bit tongue-in-cheek, the intrinsic relationship can be seen as "Riemannian integration was constructed specifically for this connection to work the way it does."

Answer 3

Reffer to method of indivisibles by Bonaventura Cavalieri (1598-1647). This method was used and learned at universities before the discovery of Riemannian integration