Determine Taylor’s integral evaluation method for K

Determine the specific computational or analytical technique used by Geoffrey I. Taylor in “The Formation of a Blast Wave by a Very Intense Explosion. I. Theoretical Discussion” (1950) to evaluate the two integrals over the similarity variable η in [0,1] that define the constant K in the energy relation E = ρ0 R^5 t^-2 K, namely the integral of ψ(η) φ(η)^2 η^2 and the integral of f(η) η^2. Clarify whether Taylor employed numerical quadrature, analytical approximations, or discrete step integration, and reproduce the procedure used to obtain his reported K values.

Background

In revisiting Taylor’s analysis, the authors compute K by numerically approximating the integrals involving the functions f(η), φ(η), and ψ(η) using Gauss–Legendre quadrature. Their computed values show minor differences compared with Taylor’s reported values.

The paper notes that Taylor’s original description indicates he evaluated the integrals using step-by-step calculations, but does not specify the method. This lack of detail prevents direct replication of Taylor’s exact procedure, motivating the need to identify the technique he used.