Reading the quote
the so called "Feynmann path integral", which, as far as I understand, means "integrating" a functional (action) on some infinite-dimentional space of configurations (fields) of a system.
leaves me wondering, how does one set up the integral of a functional such as $J[y] = \int_0^1 f(x,y,y')dx = \int_0^1 (y'^2 + xy)dx \ , \ y(0) = 0, y(1) = 1$ with E-L equation $x-2y''=0$ and solution $y(x) = \tfrac{x^3}{12} + \frac{11x}{12}$ and evaluate it explicitly? Furthermore, what does it mean to do such a thing?