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Let's think about the fundamental theorem of calculus physically.

Let's define the function v by the rule that v of t is my velocity at time t.

Physically, what does the accumulation function of velocity mean?

Well that accumulation function, remember, is say, the integral from 0 to

b, of v of t dt. And this is really the distance, that

I've traveled from time 0 to time b. This also makes sense when we think back

to Riemann sums. But what would the Riemann sum say?

if I think about how far I traveled over a short time period, say, between time

zero and time h for some small number h, right?

How far did I travel during that time period?

Well, I traveled for that small time period h and I was traveling at a speed

of, I mean, you know, a velocity at time zero, say, is a good approximation.

And I imagine my velocity didn't change very much during this time period.

So, that's a pretty good approximation for how much I traveled during the first

h moments of my journey. What about between time h and time 2h,

right? How far did I travel there?

Well, the time that elapsed was h units of time.

And how fast was I going? Well, I could use v of h.

My velocity of time h as a good approximation for my velocity over that

time period, right? My velocity is not necessarily constant.

But this is standing in for a reasonable approximation and my velocity is not

changing too rapidly. All right.

This is how far I traveled, right? Time times velocity is distance.

All right? Now how far did I travel between, say

time 2h and time 3h. Well, again, how long was I traveling for

h units of time? And how fast was I going?

v 2h is a reasonable approximation for my velocity during that time period.

And of course this keeps on going, right? But what do I get if I add all of these

things up? Right?

What I'm getting is a Riemann sum. And if I keep on adding these things up

until I get, you know, all the way to, to b, right?

What I'm writing down is the approximation for this interval of a

particular Riemann sum which approximates this integral and in the limit as h goes

to zero, that Riemann sum will compute this integral.

So, summarizing that, the accumulation function of velocity, is displacement.

And what's the derivative of displacement?

It's velocity, right? The derivative of my accumulation

function in this specific case where the accumulation function is the accumulation

function for velocity, right, the derivative of that accumulation function

is the thing that I'm integrating, velocity.

That's the fundamental theorem of calculus.

Or in symbols, you know, the accumulation function, which is[INAUDIBLE]

displacement, is the interval from zero to b of my velocity.

And what I'm asking is, how is that changing, right?

If that's my displacement when I travel from times 0 to times b, right?

What is the derivative with respect to the time that I've been traveling?

The derivative of displacement is velocity.

So physically, differentiating the accumulation function of velocity is to

differentiate displacement. It's to ask for the rate of change in my

position, which is velocity, which is the integrand.

It's the fundamental theorem of calculus just wrapped up in physical clothing.