Main content

## Word problems involving definite integrals

# Interpreting definite integral as net change

AP.CALC:

CHA‑4 (EU)

, CHA‑4.D (LO)

, CHA‑4.D.1 (EK)

, CHA‑4.D.2 (EK)

, CHA‑4.E (LO)

, CHA‑4.E.1 (EK)

## Video transcript

- [Instructor] In a previous video we started to get an
intuition for rate curves and what the area under
a rate curve represents. So, for example, this rate curve, this might represent a speed of a car and how a speed of a car is
changing with respect to time. And so this shows us that our
rate is actually changing, this isn't distance as function of time, this is rate as a function of time. So, this looks like car is accelerating. At time one it is going
ten meters per second and at time five, let's
assume that all of these are in seconds, so at five seconds, it is going 20 meters per second. So, it is is accelerating. Now the relationship between
the rate function and the area, is that if we're able
to figure out this area, then that is the change
in distance of the car. So rate or speed in this case
is distance per unit time, if we're able to figure out
the area under that curve, it will actually give us
our change in distance, from time one to time five. It won't tell us our total distance, cause we won't know what
happens before time one, if we're not concerned with that area. And the intuition for that,
it's a little bit easier if you're dealing with rectangles. But just think about this,
let's make a rectangle that looks like a pretty good
approximation for the area, let's say from time one to
time two right over here. Well what is this area from
the rectangle represent? To figure out the area we
would multiply one second, that would be the width
here, times roughly, looks like about ten meters per second. And so the units here would
be ten meters per second, times on second, or ten meters. And we know from early
physics or even before, that if you multiply a rate times time, or a speed times a time,
you're gonna get a distance. And so the unit here is in distance, as you can see, this area
is going to represent, it's gonna be an approximation
for the distance traveled. And so, if you wanted
to get an exact version, or an exact number for
the distance traveled. You would get the exact
area under the curve. And we have a notation for that. If you want the exact area
under the curve right over here, we use definite integral notation. This area right over here, we can denote as a definite integral from one to five, of R of t, dt. And once again, what does this represent? In this case when our rate is speed? This represents this
whole expression reprsents our change in distance
from t is equal to one, to t is equal to five. Now with that context, let's
actually try to do an example problem, the type that you
might see on Khan Academy. So, this right over here tells us, Eden walked at a rate of r
of t kilometers per hour, where t is the time in hours. Okay, so t is in hours. What does the integral,
the definite integral from two to three of r
of t dt equals six mean? So before I even look at these choices, this is saying, so this is
going from t equals two hours to t equals three hours and
it's essentially the area under the rate curve and here the rate is, we're talking about a speed. Eden is walking at a certain
number of kilometers per hour. So what this means,
that from time two hours to time three hours, Eden
walked an extra six kilometers. So, let's see which of
these choices match that. Eden walked six kilometers each hour? It does tell us from time two to three, Eden walked six kilometers,
but it doesn't mean, but we don't know what happened
from time zero to time one, or from time one to time two. So I would rule this out. Eden walked six kilometers in three hours. So this is a common misconception, people will look at the top bound and say okay, this area represented
by the definite integral this tells us how far in total we have walked up until that point. That is not what this represents. This represents the change in distance from time two to time three. So, I'll rule that out. Eden walked six kilometers
during the third hour. Yes, that's what we've been talking about. From time equal two hours
to time equal three hours, Eden walked six kilometers
and you could view that as the third hour, going
from time two to time three. Eden's rate increased by
six kilometers per hour between hours two and three. So let's be very clear,
this right over here, this isn't a rate, this is
the area under the rate curve, this is what this definite
integral is representing. And so this isn't telling
us about our rate changing, this is telling us how does
the thing that the rate is measuring the change
of, how does that change from time two to time three? So we would rule that out as well.