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# Analyzing problems involving definite integrals

See worked example of how to find the appropriate expression to use in order to solve word problems using definite integrals.

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• How did they even get 1200? if you plug in 2 for t in the function you get 546.6356401
• The function r(t) gives the RATE at which the population grows, not the actual population. Does this help?
• Why can't you do the integral from 0 to 7?
• Because he's asking us about the actual value of the population at t = 7 and not the net change in population from t = 0 to t = 7. To get the actual value of the population, we would add the change in population to the initial population. This is known as the Fundamental Theorem of Calculus (alternate form)

Using the Fundamental Theorem of Calculus (alternate form), we know that f(b) = f(a) + ∫ₐᵇ f(x) dx.

So, if we want to get r(7), we would do r(7) = r(2) + ∫₂⁷r(t) dt
=>
r(7) = 1,200 + ∫₂⁷r(t) dt

Takeaways:
- When asked about the net change, just get the change in quantity from and to the specified points.
• For the first question, could you also use the antiderivative from 0 to 7 of r(t) with respect to t, minus r(2)?
(1 vote)
• That would give us the population between t = 2 and t = 7, which isn't what we need. We need the total population. Hence, we add the population at t = 2 to the integral of the rate function from 2 to 7.
(1 vote)