No doubt this is the toughest thing in calculus. And when you see how long the videos below are, you'll believe me! It's the same with in-person tutoring: it always takes at least two hours to teach this stuff, usually four. It's just brutal. So don't leave all this for the last hour before your test.
What the heck are revolved solids?!
These volume problems are super-hard when it comes to actually plugging into the formulas later. But first I just want to make sure that everyone's on the same page with what the heck these "solids of revolution" things are in the first place since -- you know -- they're kind of what the whole chapter is about.
Overview: Disks vs Washers vs Shells
Hold your horses, we're not quite to the formulas yet. This video is another overview one where we'll introduce the three shapes we'll be using. It seems like most students get the hang of what disk and washer are about, but you'll definitely want to watch this if you're one of the 99% who have no clue why the heck a tube is called a shell, and how they'd help you with a volume problem.
The Disk Method
Classes always start with this one because it's the simplest, and they ain't wrong. Ultimately you'll end up always using washer method because you can use it for anything you can use disk for (just make r zero), but this is still the best way to ease into the volume pool without descending into utter confusion.
Even though washers and disks look pretty similar, they're quite a bit more complicated because now you've got two r's to think about: R and r. And you can mix them up, especially when the spin axis is to the right or above the region. In this hour-long video I take the approach I've found works the best for my tutoring clients, but I wish it could be even more plug-and-chug!
Shell Method (they should be called "tubes")
Wow, this video is worth the price of admission. Seriously, it's gonna save your bacon with shell method! The down side is it's an hour long, so no cramming. Ironically, the shell method is actually more plug-and-chug than the washer method, so hopefully after this video you'll appreciate shells more. And some of the nuances will actually help you with washers as well. Enjoy.
Disks vs Shells: picking your poison
Assuming you're not completely sick of me yet (and you're still awake), this video is shorter and takes a look at how you can decide between using disks or shells on each particular problem (assuming you're given a choice). Some regions are either way harder with one method versus the other, or can't be done at all, so I'll show you how to spot that before doing a ton of wasted work.
Volume of Swept Solids
These problems aren't quite as standard as the washer and shell problems, so your class might not cover them, or they go by a different name. Usually the words "cross section" or "sweep" or "path" will be involved, but the problem will definitely say "volume". Often they'll talk about cross sections that are triangles, squares, semicircles, rectangles, or circles.
If you do not have an account, you should get one, because it is awesome! You can save a playlist for each test or each chapter, and save your "greatest hits" into a "watch right before the final" list (not that we recommend cramming, but when in Rome...)