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## Properties of definite integrals

# Negative definite integrals

AP.CALC:

FUN‑6 (EU)

, FUN‑6.A (LO)

, FUN‑6.A.1 (EK)

## Video transcript

- [Instructor] We've already thought about what a definite integral means. If I'm taking the definite
integral from a to b of f of x dx, I can just view that as the area below my function f. So if this is my y-axis,
this is my x-axis, and y is equal to f of x,
so something like that. Y is equal to f of x. And if this is a and if this is b, I could just view this expression as being
equal to this area. But what if my function
was not above the x-axis? What if it was below the x-axis? So these are going to be equivalent. Let's say, let me just draw that scenario. So let me draw a scenario where that's my x-axis, that is my y-axis. And let's say I have, let's say I have a function
that looks like that. So that is y is equal to g of x. And let's say that this
right over here is a, and this right over here is b. And let's say that this area right over here is equal to five. Well, if I were to ask you what is the definite integral from a to b of g of x dx, what do you
think it is going to be? Well, you might be
tempted to say, hey, well, it's just the area again
between my curve and the x-axis. You might be tempted to say, hey, this is just going to be equal to five. But you have to be very careful. Because if you're looking
at the area above your curve and below your x-axis, versus below your curve
and above the x-axis, this definite integral
is actually going to be the negative of the area. Now, we'll see later on why
this will work out nicely with a whole set of
integration properties. But if you want to get
some intuition for it, let's just think about
velocity versus time graphs. So, if I, in my horizontal axis, that is time. My vertical axis, this is velocity. And velocity is going to be
measured in meters per second. Time is going to be measured in seconds. Time is measured in seconds. And actually, I'm going
to do two scenarios here. So let's say that I have a
first velocity time graph. Let's just call it v one of
t, which is equal to three. And it would be three meters
per second, so one, two, three. So it would look like that. That is v one of t. And if I were to look
at the definite integral going from time equals
one to time equals five of v sub one of t dt, what
would this be equal to? Well, here my function is above my t-axis. So I'll just go from one to five, which will be around there. And I could just think
about the area here, and this area is pretty easy to calculate. It's going to be three meters
per second times four seconds. That's my change in time. And so this is going to be 12 meters. And so this is going to be equal to 12. And one way to conceptualize this is this gives us our change in position. If my velocity is three meters per second, and since it's positive
you can conceptualize that as it's going to the right
at three meters per second. What is my change in position? Well, I would have gone
12 meters to the right. And you don't need calculus
to figure that out. Three meters per second times four seconds would be 12 meters. But what if it were the other way around? What if I had another velocity function, ;et's call that v sub two of t, that is equal to negative
two meters per second? And it's just a constant
negative two meters per second. So this is v sub two of t right over here. What would or what should the definite integral from one
to five of v sub two of t be, dt be equal to? Well, it should be equal
to my change in position. But if my velocity is negative, that means I'm moving to the left. That means my change in
position should be to the left, as opposed to to the right. And so we can just look at
this area right over here. When if you just look
at it as the rectangle, it's going to be two times
four, which is equal to eight. But you have to be very careful. Since it is below my horizontal
axis and above my function, this is going to be negative. And this should make a lot of sense. If I'm going two meters
per second to the left for four seconds, or another
way to think about it, if I'm going negative
two meters per second for four seconds, then
my change in position is going to be negative eight meters. I would have moved eight
meters to the left, if we say the convention is
negative means to the left. So the big takeaway is, if it's below your function
and above the horizontal axis, the definite integral, and
if your a is less than b, then your definite integral
is going to be positive. If your a is less than b, but your function over that interval is below the horizontal axis, then your definite integral
is going to be negative. And in the future, we'll also look at definite integrals
that are a mix of both, but that's a little bit more complicated.