5.5 Integration & Anti-derivatives
Method of Substitution
Question #5 (Medium): Word Problem
Word problems provide a main function that represent some type of a rate, and then ask to evaluate
the total amount over the time interval. Same method of substitution applies as to theoretical questions.
The key is to keep track of the units (often they are in hundreds, thousands, etc.) and make sure to
answer according to the context after the theoretical evaluation is done.
A small town rising in popularity for tourism has its population growing at the rate shown by the
function. What will the population be after three years?
, where is in years
This is a very easy function which does not require substitution.
Applying substitution would be: Let
. Rearranging, .
From now until three years later means the time interval is for the variable .
Converting the interval to fit the substitution: when ,
; and when , then
Evaluating the total population over the interval :
Therefore, the total population of the town after three years is:
, which is