1
00:00:00,025 --> 00:00:10,550
[music] Welcome back to Calculus One, and
welcome to week 11 of our time together.

2
00:00:10,550 --> 00:00:15,624
Well, this week marks a dramatic change
from what's been going on in the course

3
00:00:15,624 --> 00:00:19,926
for the last two months.
Pretty much every single week has involved

4
00:00:19,926 --> 00:00:23,440
derivatives in some form, and this week is
a sudden exception.

5
00:00:23,440 --> 00:00:27,629
We're really not going to be looking at
derivatives this week at all.

6
00:00:27,629 --> 00:00:31,879
Instead, this week, we're going to be
looking at integration and area.

7
00:00:31,879 --> 00:00:36,763
Though, you might think that what we're
going to be doing is computing a bunch of

8
00:00:36,763 --> 00:00:41,425
areas, and, and that's true, but it's be
more precise to say that this week is

9
00:00:41,425 --> 00:00:46,250
about defining the very notion of area.
Think back way at the beginning of course

10
00:00:46,250 --> 00:00:50,968
when we first introduced the derivative.
You might have said the derivative was

11
00:00:50,968 --> 00:00:55,575
computing instantaneous velocity.
But that's not the whole story, right?

12
00:00:55,576 --> 00:00:59,284
The derivative doesn't just let us compute
instantaneous velocity.

13
00:00:59,285 --> 00:01:05,687
The derivative defines what the very idea
of instantaneous velocity means.

14
00:01:05,688 --> 00:01:09,690
In the same way, this week, we're going to
cover the integral.

15
00:01:09,690 --> 00:01:14,310
And the integral does compute area, but it
computes area in the same way that the

16
00:01:14,310 --> 00:01:17,541
derivative computes instantaneous
velocity, right?

17
00:01:17,542 --> 00:01:22,970
The integral will define what area even
means.

18
00:01:22,970 --> 00:01:28,474
We're going to try and write down an
actual definition of area under the graph

19
00:01:28,474 --> 00:01:31,428
of a function.
Now, how does this fit in with the rest of

20
00:01:31,428 --> 00:01:34,716
the course?
Well, last week, in Week 10, we looked at

21
00:01:34,716 --> 00:01:39,260
anti-differentiation.
And this week, in Week 11, we're looking

22
00:01:39,260 --> 00:01:43,490
at integrals and area.
And next week, in Week 12, we'll introduce

23
00:01:43,490 --> 00:01:48,153
the fundamental theorem of Calculus.
And it's that fundamental theorem of

24
00:01:48,153 --> 00:01:52,314
Calculus in Week 12, which will tie
together the material in Week 10,

25
00:01:52,314 --> 00:01:57,351
anti-differentiation, and the material
this week, integration, and reveal that

26
00:01:57,351 --> 00:02:01,454
anti-differentiation and integration are
somehow related.

27
00:02:01,455 --> 00:02:06,158
So unfortunately, to really understand,
you know, why for instance we did this

28
00:02:06,158 --> 00:02:08,982
stuff in Week 10, you've got to wait until
Week 12.

29
00:02:08,982 --> 00:02:14,165
And this week, we're sort of stuck in the
middle where we're just defining the basic

30
00:02:14,165 --> 00:02:18,646
notions of what area means.
We're also going to be introducing sigma

31
00:02:18,646 --> 00:02:23,596
notation and doing some summation.
But that's really necessary just for being

32
00:02:23,596 --> 00:02:26,435
able to write down the definition of
integral.

33
00:02:26,436 --> 00:02:30,789
Now, I should also say that you've got a
midterm that you're probably still working

34
00:02:30,789 --> 00:02:33,676
on, and that's great.
And if you've got questions about

35
00:02:33,676 --> 00:02:37,892
anything, that midterm, the material from
this week on integration, where we're

36
00:02:37,892 --> 00:02:41,922
going, where we've been, anything about
the course at all, please contact us,

37
00:02:41,922 --> 00:02:44,325
right?
Send us a message, or write to us in the

38
00:02:44,325 --> 00:02:50,696
forum.
We want to make sure that all of us can

39
00:02:50,696 --> 00:02:57,314
make it past that finish line together.