If summation is used, it will imply that transition from healthy state to dead state will only take place at any integer time interval (i.e. in 1 year time, 2 years time etc.). This implies that deaths occurring in between integer time intervals will receive no benefit, which is not true. For example, death in 1.5 years time will still receive at end of the year in which death occurs.
Integration will include all the transitions from healthy state to dead state that take place in between integer time intervals. Hence, integration used.