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

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

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

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function.
What is the accumulation function going to

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

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

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

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

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

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

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

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

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into the accumulation function.
When is this accumulation function

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increasing?
As long as this function little f is

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positive, the accumulation function is
increasing.

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

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

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

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

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

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bigger input than there was between a and
x.

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

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accumulation function increasing.
When is this accumulation function

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decreasing?
Is it ever possible to integrate over a

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longer interval and yet get a smaller
value?

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

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

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measuring.
The integrals are not exactly measuring

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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this particular place?
Right.

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

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

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

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

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

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

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

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

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

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function's negative the accumulation
function is decreasing.

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

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

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

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accumulation functions increasing.
And another way to say that the

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

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

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

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decreasing.
The derivative of the accumulation

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function is negative.
Integrand negative.

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Derivative of accumulation function also
negative.

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

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Some sort of relationship between
derivatives.

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And integrals.