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[music] We can get a much better sense for what the integral's actually computing by

converting the integral into a function. So the way that I'm going to package

together a whole bunch of integrals is into a function, the accumulation

function. What is the accumulation function going to

do? Well, here, I've graphed y equals f of t

on the t y plane. And I picked a fixed point a, and I

imagine I've got another point that I can vary labeled x.

And the accumulation function, is the integral from a to x.

So it computes the area between a and x under the graph of this function.

And it's called the accumulation function because I imagine that as I move x to the

side, I'm kind of accumulating more and more area.

I'm determining how much area I've accumulated between a and whatever I plug

into the accumulation function. When is this accumulation function

increasing? As long as this function little f is

positive, the accumulation function is increasing.

Look, to determine whether a function's increasing, I'd have to plug in bigger

inputs and see if I get bigger outputs. That's what increasing means.

So if I were to plug in some bigger input right, the accumulation function just

figures out how much area is between a and this bigger input.

And as long as my function's positive there's now more area between a and this

bigger input than there was between a and x.

So I can summarize this in saying. That f is positive, and that makes my

accumulation function increasing. When is this accumulation function

decreasing? Is it ever possible to integrate over a

longer interval and yet get a smaller value?

Yes, it sounds counter-intuitive, but that's entirely possible.

And it really comes down to this issue about what these integrals are really

measuring. The integrals are not exactly measuring

area, they're measuring signed area. So here's an example where the function

that I'm graphing, y equals f of t, passes below the, I'm calling it here, the t

axis. And when the graph is below and I'm

calculating the integral, this area here counts as negative in the integral.

So this area which is above the axis here is calculated as honest area, but this

area here is really contributing negative area to the integral, right?

I mean, I'm taking this area and I'm subtracting this area to figure out the

integral from a to x. What it comes down to is how this integral

is defined. The integral is defined as a Riemann sum.

It's the functions value evaluated at various sample points times the widths of

those tiny rectangles and if I'm evaluating the function and the functions

value is negative then that's not really an area.

That's a signed area and it could be negative.

All right. Well in light of that, what happens in

this particular place? Right.

What happens if I plug in a bigger input to my accumulation function.

Is it possible that I could integrate over a longer interval, and yet get a smaller

value. And yeah, this is an exact picture of the

sort of situation where that happens. If I plug in a bigger input here, then

when I integrate from a to this bigger input, I'm not only subtracting this red

negative area, I'm subtracting this larger red negative area.

And in that case my accumulation function really is smaller, right?

As I take x and drag it over here. I'm gathering up more negative area, and

so my accumulation function is going down. I'll summarize that like this, if

function's negative the accumulation function is decreasing.

What does this sound like? Well what that sounds like is that the

derivative of the accumulation function is related to the thing I'm integrating, the

so called integrand. A, if I'm integrating a positive function,

accumulation functions increasing. And another way to say that the

accumulation function is increasing would be to say that the derivative is positive.

So the integrands positive, the derivative of accumulation function is positive.

The function that I'm integrating is negative, the accumulation function is

decreasing. The derivative of the accumulation

function is negative. Integrand negative.

Derivative of accumulation function also negative.

This is our first hint at the fundamental theorem of calculus.

Some sort of relationship between derivatives.

And integrals.