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A honeybee population starts with 100 bees and increases at a rate of $ n'(t) $ bees per week. What does $ \displaystyle 100 + \int^{15}_0 n'(t) \, dt $ represent?
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00:43
Frank Lin
00:47
Amrita Bhasin
Calculus 1 / AB
Chapter 5
Integrals
Section 4
Indefinite Integrals and the Net Change Theorem
Integration
Missouri State University
Oregon State University
Baylor University
Boston College
Lectures
05:53
In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.
40:35
In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.
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Okay when you look at this problem I would use the fundamental theorem of calculus. Okay. Uh to explain just this part. 0 to 15 yeah. Of N. Prime of T. D. Teeth. And the fundamental theorem of calculus says that that equals the anti derivative. It's just N. F. T. Because they undo each other and then you have the upper bound after minus the lower bound. Yeah. So what we're looking at at that piece is a, well actually I don't really care about that. Um What I'm gonna do is I'm going to add an of zero over. So what I'm deducing is that I'm writing this in green on purpose and zero plus because I'm adding it over here. The in a row from 0 to 15 of end prime of T. D. T. needs to be equal to and 15. So this I'm gonna put a circle I'm going to circle this and you might be sitting there and saying why are you already circling this is because instead of writing end of zero this is the number of bees at time zero. So no time has elapsed. It's the starting and so what I'm getting at starts with 100 is the problem didn't write and of zero here wrote 100 100 plus all of this. So that's equal to end 15. If you look at the units um what that formula gives you then is the amount of bees wow. Um Let's see because it's per week. Um That's what the 15 represents at 15 weeks. And this was a rate. So that's why this one is not a rate. So I mean circle is this is the interpretation.
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