1
00:00:03,378 --> 00:00:07,284
[MUSIC]. 
Welcome back to Calculus One. 

2
00:00:07,284 --> 00:00:10,223
And welcome to Week 15 of our time 
together. 

3
00:00:10,223 --> 00:00:14,997
We've been at this since January, and 
this is it, this is the last week of the 

4
00:00:14,997 --> 00:00:18,590
course. 
Where are we? 

5
00:00:18,590 --> 00:00:21,701
Last week in Week 14, we were working at 
techniques of integration and the 

6
00:00:21,701 --> 00:00:25,920
technique that we focused on was 
integration by parts. 

7
00:00:25,920 --> 00:00:29,690
This week in Week 15 we're looking at an 
application of integration, but which 

8
00:00:29,690 --> 00:00:32,455
application? 
There's so many applications of 

9
00:00:32,455 --> 00:00:36,410
integration that we could be looking at. 
We could fill up a whole Calculus Two 

10
00:00:36,410 --> 00:00:39,949
course with various applications of the 
integral. 

11
00:00:39,949 --> 00:00:42,705
The application that we've picked is 
volume. 

12
00:00:42,705 --> 00:00:46,695
I think it's a ton of fun, to use 
integrals to calculate the volumes of 

13
00:00:46,695 --> 00:00:51,610
solid three dimensional objects. 
It's fun because it gives us a chance to 

14
00:00:51,610 --> 00:00:54,560
think geometrically, to think 
three-dimensionally, but I also think 

15
00:00:54,560 --> 00:00:58,812
it's important historically. 
You know, thousands of years ago, the 

16
00:00:58,812 --> 00:01:02,860
first inklings of a definite integral 
appeared, you know? 

17
00:01:02,860 --> 00:01:06,552
People were thinking about taking objects 
and slicing them into really thin pieces, 

18
00:01:06,552 --> 00:01:11,390
say infinitesimal pieces, and adding up 
all of those infinitesimal pieces. 

19
00:01:11,390 --> 00:01:17,054
And the idea is really, really old. 
So, I think it's appropriate that we end 

20
00:01:17,054 --> 00:01:21,056
this course with something at the very 
beginning of humankind's grasping towards 

21
00:01:21,056 --> 00:01:25,282
the definite integral. 
We're going to end this course with some 

22
00:01:25,282 --> 00:01:28,980
volume calculations. 
It's also appropriate that we end this 

23
00:01:28,980 --> 00:01:32,452
course for the final exam. 
The final exam is available right now, 

24
00:01:32,452 --> 00:01:35,792
and you got until the beginning of May to 
finish it. 

25
00:01:35,792 --> 00:01:44,844
Good luck.