00:02
Okay, so we are going to find the volume of the region bounded by these three curves about the line y equals 10.
00:12
So i have a rough sketch of these three functions.
00:16
They're all color coded.
00:17
Notice that the region ends up being a small triangle down here, and we want to revolve it around this axis.
00:27
So you can use either washer or shell method to do so.
00:31
I'm going to use the shell method for this question.
00:35
So as a quick reminder, the volume for the shell method is 2 times pi, times the integral of the radius, times the height of each cylindrical shell.
00:49
So we can draw those on our little diagram here.
00:54
So the radius is going to go from the axis of rotation to a point inside the region.
01:06
Now, in order to do this, the vertical distance from the x -axis to the bottom point is known as just y, just a vertical distance.
01:15
So in this case, we have the full length of 10 minus the y.
01:21
So the radius is going to equal just 10 minus y.
01:27
Then the height is going to be perpendicular to the radius, and notice how it runs from the red curve to the green one.
01:43
So because it's horizontal, you're going to need to have your equations written as x equals.
01:49
So the red curve is our line 5e minus x.
01:54
This can be rewritten to be, let's see, let's see, that be x equals just 5e minus y.
02:07
And the distance for the height is going to be the right.
02:11
Point minus the left.
02:13
Or in this case, the right most point is the 5e minus y and minus the green value, which is our x equals 6.
02:24
So we will also have a minus 6 here.
02:29
And there we have our radius and height.
02:31
So we can come over to our volume integral.
02:35
We have our volume is equal to 2 pi times the integral of our radius function, which is 10 minus.
02:46
X times our 5e minus y minus 6.
02:56
Notice that because we have our integral in terms of y, we'll need a dy, and our bounds must also be y bounds.
03:08
So according to our graph, we can see that in terms of y, our bottom bound is actually the blue line, y equals five.
03:17
So that will be our bottom bound.
03:20
And the top bound is whatever y coordinate has the intersection of our red and green curves.
03:29
So in order to find the intersection, our red and green curves are here.
03:35
And notice it's just when x is equal to 6.
03:38
So you can plug in 6 in place of x and you should get 5e minus 6.
03:43
So our upper bound will be the 5e minus 6.
03:51
And now we are able to solve this integral...