Intuition for Second Fundamental Theorem of Calculus

Let's say that I have
some function, s of t,

which is positioned
as a function of time.

And let me graph a potential
s of t right over here.

We have a horizontal
axis as the time axis.

Let me just graph something.

I'll draw it kind
of parabola-looking.

Although I could
have done it general,

but just to make things a
little bit simpler for me.

So I'll draw it kind
of parabola-looking.

We call this the y-axis.

We could even call this y equals
s of t as a reasonable way

to graph our position as a
function of time function.

And now let's think
about what happens

if we want to think about the
change in position between two

times, let's say between time
a-- let's say that's time

a right over there-- and then
this right over here is time b.

So time b is right over here.

So what would be the change
in position between time a

and between time b?

Well, at time b, we
are at s of b position.

And at time a we were
at s of a position.

So the change in
position between time

a and time b-- let me write this
down-- the change in position

between-- and this
might be obvious to you,

but I'll write it
down-- between times a

and b is going to be equal
to s of b, this position,

minus this position,
minus s of a.

So nothing
earth-shattering so far.

But now let's think
about what happens

if we take the derivative of
this function right over here.

So what happens when
we take the derivative

of a position as a
function of time?

So remember, the
derivative gives us

the slope of the tangent
line at any point.

So let's say we're looking
at a point right over there,

the slope of the tangent line.

It tells us for a very
small change in t--

I'm exaggerating it visually--
for a very, very small change

in t, how much are we
changing in position?

So we write that as ds dt is
the derivative of our position

function at any given time.

So when we're talking about the
rate at which position changes

with respect to
time, what is that?

Well, that is equal to velocity.

So this is equal to velocity.

But let me write this
in different notations.

So this itself is going
to be a function of time.

So we could write this
is equal to s prime of t.

These are just
two different ways

of writing the derivative
of s with respect to t.

This makes it a
little bit clearer

that this itself is
a function of time.

And we know that this is the
exact same thing as velocity

as function of time, which
we will write as v of t.

So let's graph what v of t
might look like down here.

Let's graph it.

So let me put another
axis down here that

looks pretty close
to the original.

I'll give myself
some real estate,

so that looks pretty good.

And then let me try
to graph v of t.

So once again, if this is my
y-axis, this is my t-axis,

and I'm going to graph
y is equal to v of t.

And if this really
is a parabola,

then the slope over here is
0, the rate of change is 0,

and then it keeps increasing.

The slope gets steeper
and steeper and steeper.

And so v of t might look
something like this.

So this is the graph of
y is equal to v of t.

Now, using this
graph, let's think

if we can conceptualize
the distance, or the change

in position, between time
a and between time b.

Well, let's go back
to our Riemann sums.

Let's think about what an
area of a very small rectangle

would represent.

So let's divide this into
a bunch of rectangles.

So I'll do it fairly
large rectangles

just so we have some
space to work with.

You can imagine
much smaller ones.

And I'm going to do a
left Riemann sum here,

just because we've
done those a bunch.

But we could do it
right Riemann sum.

We could do a trapezoidal sum.

We could do anything we want.

And then we could keep going
all the way-- actually,

let me just do three right now.

Let me just do three
right over here.

And so this is actually a
very rough approximation,

but you can imagine
it might get closer.

But what is the area of
each of these rectangles

trying-- what is it
an approximation for?

Well, this one right over
here, you have f of a,

or actually I should say v of a.

So your velocity at time a is
the height right over here.

And then this distance
right over here

is a change in
time, times delta t.

So the area for that
rectangle is your velocity

at that moment times
your change in time.

What is the velocity
at that moment

times your change in time?

Well, that's going to be
your change in position.

So this will tell you--
this is an approximation

of your change in
position over this time.

Then the area of this rectangle
is another approximation

for your change in position
over the next delta t.

And then, you can imagine,
this right over here

is an approximation
for your change

in position for
the next delta t.

So if you really wanted
to figure out your change

in position between
a and b, you might

want to just do a Riemann sum
if you wanted to approximate it.

You would want to take
the sum from i equals 1

to i equals n of v of-- and
I'll do a left Riemann sum,

but once again, we
could use a midpoint.

We could do trapezoids.

We could do the
right Riemann sum.

But I'll just do a left
one, because that's

what I depicted right
here-- v of t of i minus 1.

So this would be t0, would be a.

So this is the first rectangle.

So the first rectangle, you use
the function evaluated at t0.

For the second
rectangle, you use

the function evaluated at t1.

We've done this in
multiple videos already.

And then we multiply it times
each of the changes in time.

This will be an
approximation for our total--

and let me make it
clear-- where delta t is

equal to b minus a over the
number of intervals we have.

We already know, from
many, many videos

when we looked at
Riemann sums, that this

will be an approximation
for two things.

We just talked about it'll be
an approximation for our change

in position, but it's also an
approximation for our area.

So this right over here.

So we're trying to approximate
change in position.

And this is also approximate
of the area under the curve.

So hopefully this
satisfies you that if you

are able to calculate the area
under the curve-- and actually,

this one's pretty easy,
because it's a trapezoid.

But even if this was a function,
if it was a wacky function,

it would still apply
that when you're

calculating the area under the
curve of the velocity function,

you are actually figuring
out the change in position.

These are the two things.

Well, we already
know, what could we

do to get the exact
area under the curve,

or to get the exact
change in position?

Well, we just have
a ton of rectangles.

We take the limit as
the number of rectangles

we have approaches infinity.

We take the limit as
n approaches infinity.

And as n approaches
infinity, because delta t is

b minus a divided
by n, delta t is

going to become
infinitely small.

It's going to turn into dt,
is one way to think about it.

And we already have
notation for this.

This is one way to think
about a Riemann integral.

We just use the
left Riemann sum.

Once again, we could use the
right Riemann sum, et cetera,

et cetera.

We could have used a
more general Riemann sum,

but this one will work.

So this will be equal
to the definite integral

from a to b of v of t dt.

So this right over here is
one way of saying, look,

if we want the exact area under
the curve, of the velocity

curve, which is going to be
the exact change in position

between a and b, we
can denote it this way.

It's the limit of this Riemann
sum as n approaches infinity,

or the definite integral
from a to b of v of t dt.

But what did we just figure out?

So remember, this is
the-- we could call this

the exact change in position
between times a and b.

But we already figured
out what the exact change

in position between
times a and b are.

It's this thing right over here.

And so this gets interesting.

We now have a way of evaluating
this definite integral.

Conceptually, we
knew that this was

the exact change in
position between a and b.

But we already figured
out a way to figure out

the exact change of
position between a and b.

So let me write all this down.

We have that the definite
integral between a and b

of v of t dt is equal
to s of b minus s of a

where-- let me write this in
a new color-- where s of t

is the-- we know v of t is
the derivative of s of t,

so we can say where s of t is
the antiderivative of v of t.

And this notion,
although I've written it

in a very nontraditional--
I've used position velocity--

this is the second fundamental
theorem of calculus.

And you're probably
wondering about the first.

We'll talk about that
in another video.

But this is a super
useful way of evaluating

definite integrals
and finding the area

under a curve, second
fundamental theorem

of calculus, very closely
tied to the first fundamental

theorem, which we
won't talk about now.

So why is this such a big deal?

Well, let me write it in
the more general notation,

the way that you might
be used to seeing it

in your calculus book.

It's telling us that
if we want the area

under the curve between
two x points a and b

of f of x-- and so this
is how we would denote

the area under the curve
between those two intervals.

So let me draw that
just to make it clear

what I'm talking about
in general terms.

So this right over
here could be f of x.

And we care about the area
under the curve between a and b.

If we want to find the
exact area under the curve,

we can figure it out by taking
the antiderivative of f.

And let's just say that capital
F of x is the antiderivative--

or is an antiderivative, because
you can have multiple that

are shifted by constants--
is an antiderivative of f.

Then you just have to take--
evaluate-- the antiderivative

at the endpoints and
take the difference.

So you take the endpoint first.

I guess you subtract the
antiderivative evaluated

at the starting point from
the antiderivative evaluated

at the end point.

So you get capital F of
b minus capital F of a.

So if you want to figure out
the exact area under the curve,

you take the
antiderivative of it

and evaluate that
at the endpoint,

and from that, you subtract
the starting point.

So hopefully, that makes sense.

In the next few videos,
we'll actually apply it.