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.
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.