00:03
For this problem, we are given the graph y equals our f of x equals 2 over x.
00:08
We're told we're on the interval from 4 to 8.
00:10
We want to approximate its area using m2 as two midpoint rectangle.
00:15
That's an approximation or an abbreviation that you'll see for it sometimes m2.
00:21
We don't need to draw up, but sometimes having a quick sketch makes it make a little bit more sense.
00:28
So let me go ahead and do that.
00:30
I'm interested.
00:31
So 2 over x is really the same as.
00:33
1 over x.
00:34
I know it's going to change the curve a little bit, but we're just getting a rough sketch here.
00:38
And i know that at 4, well, the height of this thing is going to be, well, it's 2 over 4, 2 over 4, which is 1⁄2.
00:46
That would be the 0 .4, 1⁄2.
00:49
And then out here at 8, where we stopped, it's 8 comma, and it's 2 over 8, which is 1 4th.
00:57
Okay, so we're trying to approximate this region in here.
01:01
We're trying to approximate the area of that thing.
01:05
So we want to use two midpoint rectangles.
01:09
So the first thing we really want to find is what is delta x? how wide is each of these rectangles? your formula for that is b minus a over n.
01:17
In our case, the b value, right? this is from a to b is our interval.
01:21
So it's 8 minus 4.
01:23
And then n is the number of rectangles, which is 2.
01:26
8 minus 4 is 4.
01:27
4 divided by 2 is 2.
01:29
They're going to be 2 units wide.
01:30
Now, i mean, that shouldn't be a surprise.
01:32
We probably could have done that one without actually going through the work here.
01:35
But sometimes it's nice to see the formula.
01:37
So you say, all right, yeah, we're going from four to eights, four units across two equal parts.
01:42
So we're going to cut it in the middle there at six.
01:45
All right.
01:46
Now, my first rectangle is going to go from four to six.
01:50
And i want to use the midpoint approximation, which means i'm going to use the height of this function at five.
01:56
So that's the height that i'm going to use here.
01:59
So that's the point five comma, and it's just two over the x value is the y value there, so two over five.
02:06
So my height is two -fiths, and my whiff is going to be the six minus four or two.
02:12
That is the whiff of each of them is my delta x.
02:16
So my first rectangle, two units wide, two -fifths of a unit tall.
02:21
So my first one is four -fifths.
02:24
For my next one, that one goes from six to eight.
02:29
Course we're going to be here at 7 and we're going to draw across using that height.
02:34
All right.
02:35
So this is 7 comma 2 over 7.
02:37
So that rectangle is going to be.
02:40
So this is my first rectangle.
02:42
It's two units wide as well, but this time it's only two sevenths units tall.
02:46
Two times two is still four over seven.
02:49
So we have to add these two together.
02:51
Let's change them into 35th.
02:53
All right.
02:54
And this guy multiplied by 7.
02:55
So this would be 28 on top.
02:57
This guy had to multiply by 5.
02:59
Five, so i have to multiply by five on top here, which would be 20.
03:05
So add those two together, and we get 48 over 35.
03:10
All right...