Adding up percentages over a continuous function

Hi,

I'm trying to figure out how to add percentages along a continuous curve--that is to say for any function f(x) where for a given x, f(x) is a percentage consumed or expended.

To make it easier to conceptualize, consider the function . Say this gives us the percentage of a birthday cake eaten by a given time t over the course of three days.

is the birthday on which the eating begins and is the end of the third day when the cake is finished. The curve reveals that the largest percentage of the cake is eaten on the birthday. A smaller portion eaten on the second, and just a few bites are left to consume on the third day.

Is there a function that can sum up the percentages over the time period so that they equal 100%?

Thanks