[MUSIC]. 
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.