00:01
We wish to determine the volume that is created when we take the area that is trapped between these four curves and revolve it around the y -axis.
00:12
First, let's get the area that's trapped.
00:16
The fourth root of x plus one has a look similar to the square root of x plus one graph.
00:24
If x is zero, it hits the y -axis at one and then goes up sharply like a square root graph.
00:35
Y equals one is a horizontal line through y equals one.
00:43
Y equals seven is a horizontal line through the y value of seven.
00:51
And x equals zero is the y -axis.
00:58
So the area that we're looking at is in here.
01:04
Now we want to revolve that around the y -axis.
01:07
If we revolve it around the y -axis, we create a look like this.
01:14
We're going to create disks and we're going to stack those disks up the y -axis.
01:28
Anytime we rotate around a vertical line, disks are stacked vertically.
01:35
Formula for the volume of the disk is pi radius squared times the height of the disk.
01:44
But the height of the disk is simply the thickness, which is a dy.
01:51
So we need to fill in the radius.
01:54
The radius is here and it's the distance from the y -axis.
01:58
We call the distance from the y -axis the x -coordinate.
02:05
The interval goes from y equals one to y equals seven.
02:12
Now since it is a dy problem, we do need to turn this x into a y.
02:18
We have y equals the fourth root of x plus one.
02:25
So let's rewrite that so that it is x in terms of y.
02:31
So we have the fourth root of x plus one equals y.
02:40
Let's subtract the one and then let's take both sides to the fourth power.
02:53
So that's going to give us x equals y minus one to the fourth power...