0:00:00.660,0:00:03.730
Let's say that I have[br]some function, s of t,

0:00:03.730,0:00:06.185
which is positioned[br]as a function of time.

0:00:12.760,0:00:15.460
And let me graph a potential[br]s of t right over here.

0:00:15.460,0:00:18.720
We have a horizontal[br]axis as the time axis.

0:00:18.720,0:00:20.140
Let me just graph something.

0:00:20.140,0:00:22.020
I'll draw it kind[br]of parabola-looking.

0:00:22.020,0:00:23.603
Although I could[br]have done it general,

0:00:23.603,0:00:26.350
but just to make things a[br]little bit simpler for me.

0:00:26.350,0:00:29.890
So I'll draw it kind[br]of parabola-looking.

0:00:29.890,0:00:31.110
We call this the y-axis.

0:00:31.110,0:00:38.350
We could even call this y equals[br]s of t as a reasonable way

0:00:38.350,0:00:42.020
to graph our position as a[br]function of time function.

0:00:42.020,0:00:44.420
And now let's think[br]about what happens

0:00:44.420,0:00:49.840
if we want to think about the[br]change in position between two

0:00:49.840,0:00:54.160
times, let's say between time[br]a-- let's say that's time

0:00:54.160,0:00:58.360
a right over there-- and then[br]this right over here is time b.

0:00:58.360,0:01:01.180
So time b is right over here.

0:01:01.180,0:01:04.540
So what would be the change[br]in position between time a

0:01:04.540,0:01:06.800
and between time b?

0:01:06.800,0:01:14.290
Well, at time b, we[br]are at s of b position.

0:01:14.290,0:01:20.380
And at time a we were[br]at s of a position.

0:01:20.380,0:01:22.570
So the change in[br]position between time

0:01:22.570,0:01:31.500
a and time b-- let me write this[br]down-- the change in position

0:01:31.500,0:01:33.400
between-- and this[br]might be obvious to you,

0:01:33.400,0:01:37.790
but I'll write it[br]down-- between times a

0:01:37.790,0:01:45.440
and b is going to be equal[br]to s of b, this position,

0:01:45.440,0:01:49.890
minus this position,[br]minus s of a.

0:01:49.890,0:01:52.757
So nothing[br]earth-shattering so far.

0:01:52.757,0:01:54.340
But now let's think[br]about what happens

0:01:54.340,0:01:58.470
if we take the derivative of[br]this function right over here.

0:01:58.470,0:02:00.320
So what happens when[br]we take the derivative

0:02:00.320,0:02:02.351
of a position as a[br]function of time?

0:02:02.351,0:02:03.850
So remember, the[br]derivative gives us

0:02:03.850,0:02:06.500
the slope of the tangent[br]line at any point.

0:02:06.500,0:02:09.430
So let's say we're looking[br]at a point right over there,

0:02:09.430,0:02:11.960
the slope of the tangent line.

0:02:11.960,0:02:14.830
It tells us for a very[br]small change in t--

0:02:14.830,0:02:18.330
I'm exaggerating it visually--[br]for a very, very small change

0:02:18.330,0:02:21.520
in t, how much are we[br]changing in position?

0:02:24.530,0:02:30.860
So we write that as ds dt is[br]the derivative of our position

0:02:30.860,0:02:32.480
function at any given time.

0:02:32.480,0:02:36.210
So when we're talking about the[br]rate at which position changes

0:02:36.210,0:02:38.850
with respect to[br]time, what is that?

0:02:38.850,0:02:41.380
Well, that is equal to velocity.

0:02:41.380,0:02:44.130
So this is equal to velocity.

0:02:44.130,0:02:46.270
But let me write this[br]in different notations.

0:02:46.270,0:02:48.450
So this itself is going[br]to be a function of time.

0:02:48.450,0:02:52.160
So we could write this[br]is equal to s prime of t.

0:02:52.160,0:02:53.540
These are just[br]two different ways

0:02:53.540,0:02:56.740
of writing the derivative[br]of s with respect to t.

0:02:56.740,0:02:58.220
This makes it a[br]little bit clearer

0:02:58.220,0:03:01.020
that this itself is[br]a function of time.

0:03:01.020,0:03:05.550
And we know that this is the[br]exact same thing as velocity

0:03:05.550,0:03:16.030
as function of time, which[br]we will write as v of t.

0:03:16.030,0:03:20.160
So let's graph what v of t[br]might look like down here.

0:03:20.160,0:03:22.270
Let's graph it.

0:03:22.270,0:03:27.140
So let me put another[br]axis down here that

0:03:27.140,0:03:29.170
looks pretty close[br]to the original.

0:03:29.170,0:03:31.230
I'll give myself[br]some real estate,

0:03:31.230,0:03:33.110
so that looks pretty good.

0:03:33.110,0:03:37.090
And then let me try[br]to graph v of t.

0:03:37.090,0:03:40.770
So once again, if this is my[br]y-axis, this is my t-axis,

0:03:40.770,0:03:43.640
and I'm going to graph[br]y is equal to v of t.

0:03:43.640,0:03:45.660
And if this really[br]is a parabola,

0:03:45.660,0:03:51.502
then the slope over here is[br]0, the rate of change is 0,

0:03:51.502,0:03:52.710
and then it keeps increasing.

0:03:52.710,0:03:54.940
The slope gets steeper[br]and steeper and steeper.

0:03:54.940,0:03:57.380
And so v of t might look[br]something like this.

0:04:02.310,0:04:08.260
So this is the graph of[br]y is equal to v of t.

0:04:08.260,0:04:10.600
Now, using this[br]graph, let's think

0:04:10.600,0:04:15.900
if we can conceptualize[br]the distance, or the change

0:04:15.900,0:04:19.990
in position, between time[br]a and between time b.

0:04:25.190,0:04:27.340
Well, let's go back[br]to our Riemann sums.

0:04:27.340,0:04:31.040
Let's think about what an[br]area of a very small rectangle

0:04:31.040,0:04:32.020
would represent.

0:04:32.020,0:04:34.570
So let's divide this into[br]a bunch of rectangles.

0:04:34.570,0:04:36.790
So I'll do it fairly[br]large rectangles

0:04:36.790,0:04:38.730
just so we have some[br]space to work with.

0:04:38.730,0:04:40.830
You can imagine[br]much smaller ones.

0:04:40.830,0:04:42.837
And I'm going to do a[br]left Riemann sum here,

0:04:42.837,0:04:44.420
just because we've[br]done those a bunch.

0:04:44.420,0:04:45.710
But we could do it[br]right Riemann sum.

0:04:45.710,0:04:47.020
We could do a trapezoidal sum.

0:04:47.020,0:04:49.030
We could do anything we want.

0:04:49.030,0:04:51.440
And then we could keep going[br]all the way-- actually,

0:04:51.440,0:04:55.560
let me just do three right now.

0:04:55.560,0:04:59.535
Let me just do three[br]right over here.

0:04:59.535,0:05:01.660
And so this is actually a[br]very rough approximation,

0:05:01.660,0:05:03.650
but you can imagine[br]it might get closer.

0:05:03.650,0:05:07.000
But what is the area of[br]each of these rectangles

0:05:07.000,0:05:10.110
trying-- what is it[br]an approximation for?

0:05:10.110,0:05:13.120
Well, this one right over[br]here, you have f of a,

0:05:13.120,0:05:15.240
or actually I should say v of a.

0:05:15.240,0:05:18.777
So your velocity at time a is[br]the height right over here.

0:05:18.777,0:05:20.360
And then this distance[br]right over here

0:05:20.360,0:05:23.250
is a change in[br]time, times delta t.

0:05:23.250,0:05:27.590
So the area for that[br]rectangle is your velocity

0:05:27.590,0:05:30.532
at that moment times[br]your change in time.

0:05:30.532,0:05:31.990
What is the velocity[br]at that moment

0:05:31.990,0:05:33.250
times your change in time?

0:05:33.250,0:05:35.650
Well, that's going to be[br]your change in position.

0:05:35.650,0:05:38.130
So this will tell you--[br]this is an approximation

0:05:38.130,0:05:41.650
of your change in[br]position over this time.

0:05:41.650,0:05:46.200
Then the area of this rectangle[br]is another approximation

0:05:46.200,0:05:50.660
for your change in position[br]over the next delta t.

0:05:50.660,0:05:52.760
And then, you can imagine,[br]this right over here

0:05:52.760,0:05:54.660
is an approximation[br]for your change

0:05:54.660,0:05:56.669
in position for[br]the next delta t.

0:05:56.669,0:05:58.710
So if you really wanted[br]to figure out your change

0:05:58.710,0:06:01.370
in position between[br]a and b, you might

0:06:01.370,0:06:04.030
want to just do a Riemann sum[br]if you wanted to approximate it.

0:06:04.030,0:06:08.480
You would want to take[br]the sum from i equals 1

0:06:08.480,0:06:12.114
to i equals n of v of-- and[br]I'll do a left Riemann sum,

0:06:12.114,0:06:13.780
but once again, we[br]could use a midpoint.

0:06:13.780,0:06:14.790
We could do trapezoids.

0:06:14.790,0:06:15.850
We could do the[br]right Riemann sum.

0:06:15.850,0:06:17.641
But I'll just do a left[br]one, because that's

0:06:17.641,0:06:24.620
what I depicted right[br]here-- v of t of i minus 1.

0:06:24.620,0:06:27.920
So this would be t0, would be a.

0:06:27.920,0:06:30.710
So this is the first rectangle.

0:06:30.710,0:06:33.956
So the first rectangle, you use[br]the function evaluated at t0.

0:06:33.956,0:06:35.330
For the second[br]rectangle, you use

0:06:35.330,0:06:37.180
the function evaluated at t1.

0:06:37.180,0:06:40.420
We've done this in[br]multiple videos already.

0:06:40.420,0:06:44.900
And then we multiply it times[br]each of the changes in time.

0:06:44.900,0:06:50.170
This will be an[br]approximation for our total--

0:06:50.170,0:06:54.900
and let me make it[br]clear-- where delta t is

0:06:54.900,0:07:00.536
equal to b minus a over the[br]number of intervals we have.

0:07:00.536,0:07:02.160
We already know, from[br]many, many videos

0:07:02.160,0:07:03.868
when we looked at[br]Riemann sums, that this

0:07:03.868,0:07:07.099
will be an approximation[br]for two things.

0:07:07.099,0:07:09.640
We just talked about it'll be[br]an approximation for our change

0:07:09.640,0:07:14.090
in position, but it's also an[br]approximation for our area.

0:07:14.090,0:07:15.270
So this right over here.

0:07:15.270,0:07:22.060
So we're trying to approximate[br]change in position.

0:07:24.720,0:07:27.820
And this is also approximate[br]of the area under the curve.

0:07:31.040,0:07:33.297
So hopefully this[br]satisfies you that if you

0:07:33.297,0:07:35.880
are able to calculate the area[br]under the curve-- and actually,

0:07:35.880,0:07:38.250
this one's pretty easy,[br]because it's a trapezoid.

0:07:38.250,0:07:41.039
But even if this was a function,[br]if it was a wacky function,

0:07:41.039,0:07:42.580
it would still apply[br]that when you're

0:07:42.580,0:07:45.630
calculating the area under the[br]curve of the velocity function,

0:07:45.630,0:07:49.400
you are actually figuring[br]out the change in position.

0:07:49.400,0:07:51.381
These are the two things.

0:07:51.381,0:07:52.880
Well, we already[br]know, what could we

0:07:52.880,0:07:57.600
do to get the exact[br]area under the curve,

0:07:57.600,0:07:59.930
or to get the exact[br]change in position?

0:07:59.930,0:08:01.720
Well, we just have[br]a ton of rectangles.

0:08:01.720,0:08:03.990
We take the limit as[br]the number of rectangles

0:08:03.990,0:08:05.940
we have approaches infinity.

0:08:05.940,0:08:08.900
We take the limit as[br]n approaches infinity.

0:08:08.900,0:08:11.960
And as n approaches[br]infinity, because delta t is

0:08:11.960,0:08:14.450
b minus a divided[br]by n, delta t is

0:08:14.450,0:08:16.100
going to become[br]infinitely small.

0:08:16.100,0:08:19.050
It's going to turn into dt,[br]is one way to think about it.

0:08:19.050,0:08:22.090
And we already have[br]notation for this.

0:08:22.090,0:08:24.760
This is one way to think[br]about a Riemann integral.

0:08:24.760,0:08:26.200
We just use the[br]left Riemann sum.

0:08:26.200,0:08:28.340
Once again, we could use the[br]right Riemann sum, et cetera,

0:08:28.340,0:08:28.840
et cetera.

0:08:28.840,0:08:30.756
We could have used a[br]more general Riemann sum,

0:08:30.756,0:08:31.810
but this one will work.

0:08:31.810,0:08:34.210
So this will be equal[br]to the definite integral

0:08:34.210,0:08:41.620
from a to b of v of t dt.

0:08:41.620,0:08:44.590
So this right over here is[br]one way of saying, look,

0:08:44.590,0:08:47.060
if we want the exact area under[br]the curve, of the velocity

0:08:47.060,0:08:49.800
curve, which is going to be[br]the exact change in position

0:08:49.800,0:08:52.120
between a and b, we[br]can denote it this way.

0:08:52.120,0:08:54.940
It's the limit of this Riemann[br]sum as n approaches infinity,

0:08:54.940,0:08:58.217
or the definite integral[br]from a to b of v of t dt.

0:08:58.217,0:08:59.550
But what did we just figure out?

0:08:59.550,0:09:02.200
So remember, this is[br]the-- we could call this

0:09:02.200,0:09:18.650
the exact change in position[br]between times a and b.

0:09:18.650,0:09:20.650
But we already figured[br]out what the exact change

0:09:20.650,0:09:22.750
in position between[br]times a and b are.

0:09:22.750,0:09:26.110
It's this thing right over here.

0:09:26.110,0:09:27.500
And so this gets interesting.

0:09:27.500,0:09:31.732
We now have a way of evaluating[br]this definite integral.

0:09:31.732,0:09:33.190
Conceptually, we[br]knew that this was

0:09:33.190,0:09:35.274
the exact change in[br]position between a and b.

0:09:35.274,0:09:37.190
But we already figured[br]out a way to figure out

0:09:37.190,0:09:39.064
the exact change of[br]position between a and b.

0:09:39.064,0:09:40.690
So let me write all this down.

0:09:40.690,0:09:43.860
We have that the definite[br]integral between a and b

0:09:43.860,0:09:55.500
of v of t dt is equal[br]to s of b minus s of a

0:09:55.500,0:10:03.150
where-- let me write this in[br]a new color-- where s of t

0:10:03.150,0:10:06.310
is the-- we know v of t is[br]the derivative of s of t,

0:10:06.310,0:10:17.147
so we can say where s of t is[br]the antiderivative of v of t.

0:10:17.147,0:10:18.855
And this notion,[br]although I've written it

0:10:18.855,0:10:21.950
in a very nontraditional--[br]I've used position velocity--

0:10:21.950,0:10:24.245
this is the second fundamental[br]theorem of calculus.

0:10:26.984,0:10:28.900
And you're probably[br]wondering about the first.

0:10:28.900,0:10:30.810
We'll talk about that[br]in another video.

0:10:30.810,0:10:33.460
But this is a super[br]useful way of evaluating

0:10:33.460,0:10:35.620
definite integrals[br]and finding the area

0:10:35.620,0:10:38.780
under a curve, second[br]fundamental theorem

0:10:38.780,0:10:42.660
of calculus, very closely[br]tied to the first fundamental

0:10:42.660,0:10:44.530
theorem, which we[br]won't talk about now.

0:10:44.530,0:10:46.600
So why is this such a big deal?

0:10:46.600,0:10:48.970
Well, let me write it in[br]the more general notation,

0:10:48.970,0:10:50.761
the way that you might[br]be used to seeing it

0:10:50.761,0:10:51.960
in your calculus book.

0:10:51.960,0:10:53.780
It's telling us that[br]if we want the area

0:10:53.780,0:10:58.260
under the curve between[br]two x points a and b

0:10:58.260,0:11:01.710
of f of x-- and so this[br]is how we would denote

0:11:01.710,0:11:04.550
the area under the curve[br]between those two intervals.

0:11:04.550,0:11:06.540
So let me draw that[br]just to make it clear

0:11:06.540,0:11:09.270
what I'm talking about[br]in general terms.

0:11:09.270,0:11:12.050
So this right over[br]here could be f of x.

0:11:12.050,0:11:15.710
And we care about the area[br]under the curve between a and b.

0:11:15.710,0:11:19.570
If we want to find the[br]exact area under the curve,

0:11:19.570,0:11:24.690
we can figure it out by taking[br]the antiderivative of f.

0:11:24.690,0:11:32.780
And let's just say that capital[br]F of x is the antiderivative--

0:11:32.780,0:11:36.095
or is an antiderivative, because[br]you can have multiple that

0:11:36.095,0:11:43.916
are shifted by constants--[br]is an antiderivative of f.

0:11:43.916,0:11:47.330
Then you just have to take--[br]evaluate-- the antiderivative

0:11:47.330,0:11:49.440
at the endpoints and[br]take the difference.

0:11:49.440,0:11:52.080
So you take the endpoint first.

0:11:52.080,0:11:56.230
I guess you subtract the[br]antiderivative evaluated

0:11:56.230,0:12:00.200
at the starting point from[br]the antiderivative evaluated

0:12:00.200,0:12:01.090
at the end point.

0:12:01.090,0:12:07.871
So you get capital F of[br]b minus capital F of a.

0:12:07.871,0:12:10.370
So if you want to figure out[br]the exact area under the curve,

0:12:10.370,0:12:13.030
you take the[br]antiderivative of it

0:12:13.030,0:12:16.050
and evaluate that[br]at the endpoint,

0:12:16.050,0:12:18.840
and from that, you subtract[br]the starting point.

0:12:18.840,0:12:20.340
So hopefully, that makes sense.

0:12:20.340,0:12:23.530
In the next few videos,[br]we'll actually apply it.