00:01
So here we're going to look at a solid of revolution.
00:07
And i actually follow some steps when i'm working on these.
00:12
The first step that i do is visualize the object that i'm trying to create with this revolution.
00:23
And so usually there's some information given that may not be too simple to interpret.
00:28
I think there may be a little bit of.
00:31
Of uncertainty in the problem that's given, but i've interpreted this following way.
00:39
There's no problem that we're doing a solid that's created by the curve, y equals x to the 6, which is kind of a very quickly curving upwards function.
00:54
And it is bounded by y equals positive 1.
00:58
And i think what we're doing is making a revolution about the y equals minus four line.
01:06
So that's the interpretation that i'm doing.
01:10
In any event, something similar is probably showing up in that problem.
01:18
And so that's my first step, is that we are going to be taking that green region and swooping it around the red line.
01:28
And it's going to make kind of a weird shaped donut type thing.
01:32
I don't even know how to draw that, but something with a bit of a sloped side.
01:39
Can't see the internal part, but it will go down to a height of zero in the middle.
01:49
That'll look quite right.
01:53
So it'll go down to zero and then kind of, but it will fill up with some stuff.
02:00
So it'll look kind of weird like maybe a weird donut.
02:06
Let's put it that way with a hole in the middle.
02:09
Okay, so something like that.
02:13
And then my next step is to figure out a strategy.
02:20
There are really only two that you use with volumes of revolution.
02:24
One is called a disk method, and the other is called the shell method.
02:31
And i see i'm going to have a big gap, like hole -like thing.
02:36
And so it's usually easier to do what's called the cylindrical shell.
02:40
I'll draw what we're doing a little bit.
02:44
So that's the method we're going to use.
02:47
So the idea is to break up that green region into small strips.
02:54
And imagine each strip making a cylinder that has a certain radius and a certain length and a certain thickness, delta r.
03:13
And the volume is the circumference 2 pi r times the length, but that's kind of like the surface area, but then there's a small thickness delta r.
03:28
And here i see what my radius is for that shell.
03:32
It's going to come up from the y equals 4, y equals minus 4.
03:39
So there's a radius of 4 and then y all the way up to the top there...