Summary: This paper considers the non-parametric estimation of the integral \[ J=\int^{T}_{0}t \lambda_ 2(t) \phi (t) dt, \] where \(\lambda_ 2(t)\) is the unknown second-order intensity function of a two-dimensional stationary isotropic point process observed in some region and \(\phi\) (t) is known for \(t\in [0,T]\). An unbiased estimator of J is derived, and a computionally fast approximation to it is proposed. The estimator is then used to obtain a kernel method for smoothing point process data, a new estimator of the Fourier transform of the second-order intensity and some tests for spatial association between a point process and another stochastic process.