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We've seen what happens when we differentiate the accumulation function.

That's the fundamental theorem of calculus.

If I take the derivative of the integral from a to b of f of x dx, well what do I

get? Well, it's just f.

Of b, right? The rate of the change of the

accumulation function is the functions value.

But, what would happen if I were to differentiate with respect to the left

hand end point instead of with respect to the right hand end point?

By which I mean. What's the derivative with respect to a

of the integral from a to b of f of x dx. Right.

What happens if instead of diferentiating with respect to the top end point, I'm

diferentiating with respect to the bottom end point?

I'm asking how is this function as a function of a changing?

Let's think about the graph. Let's draw my coordinate axes.

And I'll draw some random looking function.

And I'll pick some points a and b. And the integral from a to b calculates

the area in here. And I want to know how does that integral

change when I wiggle a? I'm asking to differentiate the integral

with respect to this. Left-hand endpoint.

So let's wiggle the left-hand endpoint. Let's move it over a little bit to a plus

h. I'm imagining h is very small.

Alright, and I want to know, how does the integral change?

Well the quantity that calculates the absolute change in the integral is this.

What's this thing here? This is the integral from a plus h to b.

And this thing here is the integral from a to b, so this difference is telling me

how the integral changes when I replace a by a plus h.

Now if you think about it, what this is really calculating is, you know, related

to the integral from just a plus h, right.

This integral is calculating this area, and subtracting this larger area.

So the difference is really just this area in here between A and A+H but it

comes with a negative sign because I'm subtracting this smaller area and I'm

subtracting now this larger area. So I've got this negative sign here.

Now this region, if h is small enough, is practically a rectangle, and it's

practically a rectangle of width h and height let's say f of a.

So, that means that this difference is at least approximately just h times the

function's value at a. Now how does that help?

Remember what I'm trying to calculate. I, I'm trying to differentiate the

interval from a to be with respect to a. That means I'm trying to take the limit

of this difference quotient. But the numerator here, we just saw, is

approximately, negative h times f of a. And that means in the limit I expect to

get an answer of just negative f of a, alright, these hs will cancel in the

limit, and that's exactly what I hope for, right?

The derivitive of the integral from a to b with respect to the left hand endpoint

a is negative f of a. So I can summarize this.

So I can summarize this. The derivative with respect to a of the

integral from a to b of f of x dx is negative f of a.

This fact coheres with a certain convention about integration.

The convention that we use is that if we integrate from a to b.

the function f of x dx that's negative the interval from b to a of f of x dx.

So if you integrate the wrong way, so to speak, we want to count that as negative

area.

>> Now in light of this convention, what do we know?

>> So d da of the integral from a to b of f

of x dx is d da of this, negative the integral from b to a of f of x dx.

But now this is the derivative of the top end point which is just the usual

fundamental theorem of calculus. So this is negative f of a.

So the upshot here is that differentiating the integral from a to b,

with respect to the left hand or bottom end point, Is negative the function's

value. And that make sense because if you

increase the left hand endpoint that decreases the area.

Alright, so it seems reasonable that a negative sign should be popping up there.

And the cool thing is that, that fact is cohering with a convention about

integration. That compared to integrating the usual

way, if you integrate the wrong way you introduce a negative sign.