Geometric approaches to the properties of the definite integral: enhancing understanding through visual demonstrations

From a purely geometric perspective, this paper provides demonstrations of two properties related to the definite integral. The first one discusses the theorem of the definite integral of a function and its inverse. The second addresses the calculation of the area of a region rotated around the line y = x. The intention is to illustrate that when feasible, it is advantageous to explore the geometric aspects of the elements involved in a property for the purpose of its demonstration. In both cases, we assume only the monotonicity and continuity of the function.