The perturbation of a thermal neutron flux field by an absorbing foil is considered for the case of a foil of thickness t and of lateral dimensions ≫ L, where L is the diffusion length of thermal neutrons in the medium. The integral equation for “one-velocity” transport of neutrons in the medium containing the foil is solved by a variational method in which the “eigenvalue” is closely related to the foil activation. The results are compared with the predictions of the Bothe and Skyrme theories. The Bothe and Skyrme theories are compared for the case of the finite disk-shaped foil and are shown to differ primarily in the transport correction. This difference may be important in cases where L is not very large compared with the mean free path of neutrons in the medium. On the basis of these considerations, a new analytic approximation for the activation of a finite foil is proposed.