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Introduction
Published in Chandrakant S. Desai, Tribikram Kundu, Introductory Finite Element Method, 2017
Chandrakant S. Desai, Tribikram Kundu
The idea is analogous to what Eudoxus and Archimedes called the method of exhaustion. This concept was used to find areas bounded by curves; the available space was filled with simpler figures whose areas could be easily calculated. Archimedes employed the method of exhaustion for the parabola (Figure 1.7); here by inscribing an infinite sequence of smaller and smaller triangles, one can find the exact numerical formula for the parabola. Indeed, an active practitioner of the finite element method soon discovers that the pursuit of convergence of a numerical procedure is indeed fraught with exhaustion!
Representational fluency in calculating volume: an investigation of students’ conceptions of the definite integral
Published in International Journal of Mathematical Education in Science and Technology, 2022
The chopping up method is another conception for the definite integral, which ‘means the classical Greek method of exhaustion refined later to the Riemann sum approach’ (Czarnocha et al., 2001, p. 99). With reference to Oehrtman’s (2009) idea of collapsing, Jones and Dorko (2015) explained this conception and how it differs from the indivisibles and MBS conceptions as: The ‘chopping up method’ conception is identical to the Riemann integral in that a finite partition is created, which is successively refined until the limit is reached. It is different from MBS in that MBS already takes as assumed that the pieces are infinitesimally small in size. It is also different from indivisibles since the refined partition never collapses in dimension. (p. 156)These conceptions were purposefully chosen from the literature because of their convenience for interpretations of definite integrals that students might utilize while calculating the volume of a solid of revolution.