Calculus in the ring of Fermat reals, Part I: Integral calculus

• We develop integral calculus for smooth functions on Fermat reals, based on the existence and uniqueness of primitives.
• The total order relation permits to prove several classical order properties for integrals and to study multiple integrals.
• In the quasi-topos of Fermat spaces, we prove smoothness of integral operators between infinite-dimensional spaces.
• The use of nilpotent infinitesimals permits to simplify calculations and to formalize informal approaches in physics.
• This approach to infinitesimals does not require a background in mathematical logic and is compatible with classical logic.

We develop integral calculus for quasi-standard smooth functions defined on the ring of Fermat reals. The approach is by proving the existence and uniqueness of primitives. Besides the classical integral formulas, we show the flexibility of the Cartesian closed framework of Fermat spaces to deal with infinite-dimensional integral operators. The total order relation between scalars permits to prove several classical order properties of these integrals and to study multiple integrals on Peano–Jordan-like integration domains.