The process known as the differentiation of functions in the Calculus does not present the insurmountable difficulties one experiences in attacking the converse problem associated with the Integral Calculus. As is well known, there are many functions which we cannot integrate, and as the process of integration is really a summation, this means there are many areas which cannot be evaluated by exact methods. There are many cases in various branches of science in which it is required to determine the area of a closed curve obtained by plotting a series of observations. The engineer, for example, has to determine the areas of indicator and similar diagrams, the naval architect has to calculate volumes, and centres of gravity or buoyancy by approximate methods. The mathematician may be called upon to evaluate definite integrals which cannot be integrated directly, such as the elliptic integrals. The problem reduces simply to the consideration of the determination of the area between the curve y=f(x), the ordinates x=x0 and x =x0+nh, and the axis of x; and the first part of the paper will be devoted to the methods of obtaining empirical rules for approximation to the evaluation of such an area when f(x) cannot be directly integrated. In the second part, the mechanical methods of integration will be considered, including demonstrations with the different types of instruments illustrating these methods.