00:04
Okay, we are asked to revolve the region bounded by these graphs about the y -axis.
00:11
We're told to use shells, and certainly that's going to make sense for us here.
00:14
Let's get a quick sketch of what this looks like.
00:18
X plus this, well, if you think about e to the negative 1 over x squared, or that would be 1 over e to the x squared, you know, it's not doing too much besides for values right around here.
00:29
You're just going to get something that sort of looks like y equals x, and then it's going to bubble up a little bit here, but it's not going to be a very exciting graph to look at.
00:38
You can always check these things with a graphic calculator, decimals, however, you're allowed to sketch them.
00:43
But we just want to make sure what function is the top, which one is the bottom, or left and right sometimes.
00:49
So that's what that function is going to look like.
00:52
Y equals zero we know is just the x axis, x equaling zero is the y axis, and then x equaling one.
00:59
So, you know, the stuff bounded by those regions would be this.
01:03
Shape right in here.
01:05
So that's what we, that's the region that is bounded.
01:08
And now we're going to revolve that and we wanted to go about the y -axis.
01:13
Okay.
01:14
And now it did tell us to use shells and that's good because i wouldn't want to do this one sideways.
01:19
I wouldn't want to do this one using disks and washers simply because the right function and left function would change right about here, and it would be kind of an annoying thing to have to work with.
01:32
All right, so we're going to use shells instead.
01:34
So for shells, basically, we have to know our formula for it, and really all it is is finding the lateral surface area of a cylinder, which is 2 pi rh.
01:45
All right, so that's the lateral surface area of the cylinder, because that's what i'm doing is i'm taking this rectangle, and i am literally just revolving it about that axis and making a solid out of that shape.
01:57
So that one rectangle would give me this shell, this cylinder.
02:01
And then i do the one next to it, and i get another one, and then around it, and we make that solid shape.
02:06
All right.
02:06
So your formula is 2 pi times the integral of r times h.
02:12
Now, because my rectangle is vertical here, since it's going up and down, i need to have everything in terms of x.
02:19
So i get my first point to my last point.
02:22
So i need these guys in terms of x as well, and i need my x limits.
02:27
All right, so it is 2 pi times the integral.
02:31
Now, x limits are pretty simple.
02:33
It's going to be from 0 to 1.
02:34
They basically told us that...