 So let's take a look at another example of the definite integral. So here's a fairly standard demographic problem. What I have is the natural rate of increase of a country. That's the number of births minus the number of deaths. And it roughly corresponds to the population growth of a country if we discount the possibility of immigration and emigration. And I have some function that gives that natural rate of increase. And I want to know how the population will change between 2020 and 2050. So again, we might consider a little bit of dimensional analysis to tell us what we should be looking at. And again, I might start off, T is measured in years. Now if you have some questions about whether or not this quantity is relevant, we might play Jeopardy! Which is to say, let's start off with an answer in this quantity and see if it is an answer to our question. So the answer is 30 years. The question is, how will the population change? And as an answer to the question, how will the population change, 30 years doesn't seem very good. It seems like it's an answer to how long will it take or maybe how much older is the population or something like that. But it doesn't seem that T is going to be relevant. So I'm going to ignore it. How about N of T? This is measured in persons per year, millions of persons per year. And so I might get an answer like 27 million persons per year. And so we play Jeopardy! 27 million persons per year is the answer. Is the question, how will the population change? Now there's a subtle point here. It seems like that might be an answer to that question. But it isn't. 27 million persons per year would be an answer to the question how rapidly will the population change or what rate will the population change by. But in a question like how will the population change, it doesn't seem like millions of persons per year is going to be an answer. So we'll deem this to be not relevant. And then finally let's take a look at our integral. This is going to be measured in persons. And so I have the Jeopardy! answer 37 million persons. The question is how will the population change? And as an answer to the question how will the population change, 37 million persons, I can accept that. That seems to be a plausible answer. So it looks like we're going to be looking at the definite integral. And so we'll set that up. So our population changes our definite integral of N of T. T is controlling. So I need to find T values. Well 2020, T is years after 2000. So 2020 is 20 years after. 2050 is 50 years after. And so the definite integral I'm interested in looks like that. And so I find an anti-derivative. And I evaluate that at the two endpoints. And after all the dos-settles I get an answer 75.610. Don't forget the units here. Are going to be in millions of persons. Millions of persons per year times years. Gives us our answer in millions of persons. And as an answer, again final Jeopardy! 75.610 million persons. Question is how will the population change? And that seems to be a good answer to that question.