00:02
In this question, we are asked to find the volume enclosed by this plane and the coordinate planes in 3 -space.
00:17
Specifically, this region is contained in the first octant, where all of x, y, and z are positive.
00:27
Now, in order to get started, let's draw a diagram of what this looks like.
00:33
So, first of all, we have our three coordinate axes that define this 3 -space.
00:47
And the easiest way to represent a plane like this on this type of representation is by its intercepts with the axes.
01:02
So if we set y and z equal to 0, then we'll find a spot on the x -axis, that is, the x -intercept.
01:14
By inspection, that would be 4.
01:17
The same thing is true if we set x and y equal to 0.
01:30
The z -intercept is 4.
01:33
And then if we set x and z equal to 0, we have 2y equals 4, which means y equals 2.
01:42
And so our plane intercepts the coordinate planes in these three straight lines.
01:55
And the region we have is the one contained within that, with this surface, this surface, and these two surfaces.
02:08
Okay, now we're ready to calculate the volume.
02:20
As we may know, the volume can be written as the triple integral of 1.
02:34
To evaluate it, we have to choose an order of integration and determine the limits for that particular order.
02:43
Now, because we are talking about a region enclosed by planes, where only one of these planes is not a coordinate plane, each of the orders will be equivalent.
03:00
That is, when we integrate with respect to 1, we will be integrating from a coordinate plane to the slanted plane.
03:16
So for example, that's what this would look like from the side, for the z and y -axes.
03:22
And then we would integrate over a triangle perpendicular, in the plane perpendicular to the axis of the first variable that we integrated.
03:39
And so i will choose the order z first, then y, and then x.
03:51
So in that case, i need ranges for z, first of all, bounds for z, that apply to all x and y values.
04:06
Now i know my lower bound is the xy -plane.
04:09
That's the collection of points in my region that has the lowest z -coordinate.
04:19
The highest z -coordinates for each pair xy, or i should say the highest z -coordinate for each pair, is the one on this slanted plane.
04:41
And so i can say z ranges between 0 and that plane, expressed as a function of x and y.
04:50
Then once we've integrated with respect to z, we have the general expression for an integral along a vertical line in this diagram.
05:13
And then so we just have to integrate along this direction and this direction, since with respect to x and y, this line is just a point...