Linear Approximation Examples
In this video we'll work more examples (you're working these along with me, right?), as well as talking about error and percent error.
Author Archives: hangtime
What's A Linear Approximation
In this video I explain what the heck we're talking about with linear approximations, and show you where the formulas come from while working a couple of examples. If you're not hot on tangent lines, definitely brush up on the tangent lines chapter, which also reviews the slope-intercept form of lines.
Using the equation of the tangent line to approximate values of functions.
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.
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.
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.
Washer Method
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!
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.
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.
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.
Volume: the hardest topic in calculus. Four hours of videos get you through the integral disk method, washer method, and shell method.
How To Decide Between dx & dy
In this video we pick up where we left off in the previous video, including working a problem in both dx and dy to see the difference. I also discuss how these problems relate to the revolved solids coming up in the next chapter, which are the most difficult topic in all of calculus for most students!
Finding Area Using dy
This video starts with problems that are "obvious" dy problems -- meaning it's pretty straightforward that you need to use dy -- but then it moves into problems which are tougher to figure out whether to use dx or dy. By the end of this video, you'll be able to relax and let the problem tell you whether it wants to be dx or dy.
Finding Area Using dx
When it comes to dx vs dy, students I've worked with ALWAYS prefer dx. It's just what everyone has been using for functions since they were introduced in Algebra 2. So we'll start with area problems involving dx (and thus functions of x) because it's the easiest place to start from.
Using dx and dy integrals to find the area between functions
Improper Integrals: Upper or Lower Bound Is Infinity
Finally, a problem that involves both limits AND integration! Your favorite! I don't like these either, but at least they're easy to spot: they're the ones with infinity in a definite integral. Students tend to make tons of mistakes on these (as do I), but if you practice these on your homework, you too can plug-and-chug through them like a champ.
Improper Integrals with Discontinuities
These ones are hard to spot because they don't have an infinity in them. But then again, limits are really just plugging in anyway, so if you don't notice it's an improper integral, there's still a decent chance you'll get the right answer!
Integrals where infinity is one of your limits of integration, or the function doesn't exist at one of the limits.
Second Fundamental Theorem of Calculus
If your teacher is covering these, you'll see a big "Fundamental Theorem of Calculus" in your syllabus. Probably. These are weird because they involve taking the derivative of an integral, but that actually makes them twice as easy instead of twice as hard.