formula for tpx in terms of hazard rate function

[burtonnm]

There are two formulas for tpx in terms of the hazard rate function. The only difference is the variable of integration. How do you tell which formula to use and in each case what are the limits of integration?

If I were to take a shot at answering this myself I would say if the hazard rate function is given for a specific age then you integrate from 0 to t (nondeferred). If the hazard rate function is given in terms of the variable x (with no specific age given) then you integrate from x to x+t. Is this right?