[MUSIC]. 
Welcome back to Calculus One. 
And welcome to Week 15 of our time 
together. 
We've been at this since January, and 
this is it, this is the last week of the 
course. 
Where are we? 
Last week in Week 14, we were working at 
techniques of integration and the 
technique that we focused on was 
integration by parts. 
This week in Week 15 we're looking at an 
application of integration, but which 
application? 
There's so many applications of 
integration that we could be looking at. 
We could fill up a whole Calculus Two 
course with various applications of the 
integral. 
The application that we've picked is 
volume. 
I think it's a ton of fun, to use 
integrals to calculate the volumes of 
solid three dimensional objects. 
It's fun because it gives us a chance to 
think geometrically, to think 
three-dimensionally, but I also think 
it's important historically. 
You know, thousands of years ago, the 
first inklings of a definite integral 
appeared, you know? 
People were thinking about taking objects 
and slicing them into really thin pieces, 
say infinitesimal pieces, and adding up 
all of those infinitesimal pieces. 
And the idea is really, really old. 
So, I think it's appropriate that we end 
this course with something at the very 
beginning of humankind's grasping towards 
the definite integral. 
We're going to end this course with some 
volume calculations. 
It's also appropriate that we end this 
course for the final exam. 
The final exam is available right now, 
and you got until the beginning of May to 
finish it. 
Good luck.