2.3 Volumes of Revolution: Cylindrical Shells

[STARTS AT 9:14]

PROFESSOR: So define Q as the region bounded on the right by g of y equals 3/y and the left by the y-axis over the interval from 1 to 3. Well, based on this description, it matches almost word for word our rule, the shell method with revolution about the x-axis. So I'm pretty sure it's going to fit, but I still want to go ahead and sketch this out. 

So 3/y actually looks like this. I'm kind of sketching it. But at 1, it is 3. And at 3, it's 1. And really that's at a y-value of 1, the x-value, the output, is 3. At a y-value of 3, the output is 1. So I have those points that line up. 

I am revolving this around the x-axis. So my representative rectangle looks like this. Now, again, this matches are set-up. We're about the x-axis. So my volume is going to be equal to the integral from 1 to 3 2pi y and then g of y, which is 3/y, dy. 

So that would be the integral. And the 3 will go ahead and factor out the 2pi. And that is going to be 3. So 2pi 3y evaluated from 1 to 3. So that'll 9-- so 18 pi minus 6pi. So that is 12pi units cubed. 

Now, again, since this was in terms of y, that-- this is just what made sense to do. And, generally, that will be the case. But that's not always the case.

[ENDS AT 11:36]