00:01
So we're dealing with the graph of y equals x cubed, x equals 0, and x equals 3.
00:11
At least that's what the numerade says to write down.
00:14
I'm not sure if this is x -tube -sosed to be y -equals to be y -equals 3, but i can't open up the book.
00:18
And we want to rotate this around the y -axis.
00:21
And so around the y -axis, there are actually two possibilities to use a shell method and also to use the disk method.
00:32
And so let me draw.
00:35
We have y equals x cubed.
00:36
It's going to look something like this.
00:39
And x equals zero is this line.
00:43
And x equals three.
00:45
One, two, three is this line.
00:47
In fact, why don't i do one, two, three so that i can get this up here.
00:52
Try to go up straight.
00:54
And that would intersect there at the point three and 27.
00:58
And i believe this is the region that they're discussing rotating around this axis.
01:05
So i'm just going to kind of symmetrically draw this other part over here.
01:10
And so i believe that's what the question is asking.
01:14
And so let's use the disk method.
01:17
And the disc method, we're going to think of having a disc that is an outer disc and that are the washer method.
01:24
And that little ring is going to be real thin.
01:27
And it's going to be d .y thin.
01:30
So we're going to use d .y method.
01:32
So our integral would be written in terms of y.
01:36
And so we're going to let y be as small as zero and go as big as 27.
01:40
So zero to 27.
01:43
And that outer ring is always having a radius of three.
01:48
So it has an area of pi times three squared.
01:53
Now we need to subtract away that inner radius.
01:56
And we need that inner radius to be written in terms of x equals.
02:01
So x is equal to y to the one -third power.
02:04
So that's this radius.
02:07
And so we have pi times y to the one -third squared...