In this paper, we prove the existence, uniqueness and stability of the solution of an integral geometry problem (IGP) for a family of curves of given curvature. The functions in the statement of the curvature depend on two variables, which is occured especially in the case of IGP along geodesics. To prove the solvability of the problem, we reduce the IGP to an over determined inverse problem for the transport equation. We also develop a new symbolic algorithm to compute the approximate solution of the problem and present two computational experiments to show the accuracy of the algorithm. The results show that the proposed approach provides highly accurate solutions and it is robust against data noises.