00:01
For this problem, we want to estimate the volume of the solid that lies below the curve given here as z.
00:08
And we are going to be looking at the area from x equaling 1 to 2 and from y equaling 0 to 3.
00:17
And we're going to be approximating this using riemann sums with m and n both equal to 2.
00:23
So let's set this setup.
00:25
What does this look like? well, one, two, three, one, two, three.
00:32
Okay, so i'm looking at this piece right here, x going from one to two, and zero going one to three.
00:40
So that's the base, that's r.
00:44
That's where the blocks are going to be sitting, and then they're going to be standing up as high up as our z curve.
00:50
Now, since m and n are both two, that means i'm going to be dividing both pieces into two.
00:59
So i've got two, four pieces in total.
01:02
So what i'm going to do here? to start with, we are going to be taking the height.
01:08
We're going to use the lower left corners.
01:11
And i'm just going to put a little dot here just to kind of remind us, lower left corner.
01:16
Whatever the value of the function is at those points, of those lower left corners, that will be the height of our blocks.
01:22
And if you look at the base, every block here is the same size.
01:26
And each block has the same base.
01:27
So the base of one of our blocks is going to be one half.
01:31
That's the distance along the x -axis.
01:34
And one and a half the distance along the y -axis.
01:37
So volume is that base times the height.
01:41
So we're going to add up the heights.
01:44
There's going to be four of them.
01:45
In order to do this, i'm going to make a little chart.
01:48
I'm going to have x, y, and then z, which is the height.
01:53
That tells me how high up i have to go to reach that curve.
01:56
So my x's will be one and one and a half.
02:03
And so i've got two of those...