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

Â That's the fundamental theorem of calculus.

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

Â get? Well, it's just f.

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

Â accumulation function is the functions value.

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

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

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

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

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

Â diferentiating with respect to the bottom end point?

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

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

Â And I'll draw some random looking function.

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

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

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

Â with respect to this. Left-hand endpoint.

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

Â h. I'm imagining h is very small.

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

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

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

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

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

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

Â to the integral from just a plus h, right.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â This fact coheres with a certain convention about integration.

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

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

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

Â area.

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

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

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

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

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

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

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

Â value. And that make sense because if you

Â increase the left hand endpoint that decreases the area.

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

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

Â integration. That compared to integrating the usual

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

Â