Abstract: The concepts of the lower approximation integral，the upper approximation integral and rough integrals are given
on the basis of function rough sets.With these concepts，double median theorem of rough integrals and the relationships of
the lower approximation integrals，the upper approximation integrals and rough integrals are discussed.Rough integrals also
have finite shrinkage characteristic and finite extension characteristic.The rough degree and precision degree of function
rough sets are presented，and the relationships of these are gotten.Based on above of all function rough sets’choice-surplus
theorem is obtained.

摘要： 在函数粗集的基础上给出了下近似积分、上近似积分与粗积分的概念，利用这些概念不仅给出了粗积分的可分辨关系、不可分辨关系、有限萎缩性和有限扩张性定理，还定义了函数粗集的精度与粗糙度及二者之间的一些关系，并给出函数粗集的筛选-剩余原则。