Pg 15 of the ActEd notes (Chapter 7) - the 2nd last line where the bounds of the integral are being evaluated

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The expression in the square bracket [ (t) x (tpx) ] evaluated at t=0 is zero.

And the expression [ (t) x (tpx) ] evaluated at t=(w-x) is zero for the following reasons:

- The probability that a life aged x survives for at least (w-x) years is 0
- w is defined as the maximum age of an individual i.e. it is assumed the individual will not live past the age w (it is the limiting age)
- (w-x) is the maximum number of years a life could still live from its current age x
- so if a life has already survived x years, the probability that it survives
**at least**a further (w-x) years would take the individual past its limiting age (try drawing a timeline to illustrate the concept) -
in other words, the probability that a life now aged x years old is
**still alive**after (w-x) years is 0

Drawing a timeline can be quite useful when working with survival probabilities!

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