00:01
In this problem, we use n equals 4 to estimate the area under the graph of the given function with the trapezoidal rule, the midpoint rule, and simpson's rule.
00:11
So we start by calculating delta x.
00:13
We're going from x equals 0 to x equals 4.
00:16
So we have 4 minus 0 divided by n, in this case, 4.
00:20
So delta x is 1.
00:24
Well, that allows us to go to the graph and estimate the corresponding y coordinates.
00:29
We're going to partition up the interval from 0 to 4 using that delta x.
00:36
So let's come over here.
00:38
So here's our x.
00:39
Here's our y or our f of x coordinates corresponding to that.
00:43
So what does our partition look like? x naught starts with the left endpoint, so it is a 0.
00:50
And we will simply add a delta x, working our way over.
00:55
And we should hit, whoops, got it carried away there.
00:59
That's a 2.
01:04
What we should get when we get to x sub 4, x sub n, we should get the right endpoint.
01:09
There we go.
01:11
All right.
01:11
You can look at the graph, and here are the estimates that i see for the y coordinates.
01:17
Yours may be slightly different.
01:19
But what i see based on the graph is that when x is 0, y is 0.
01:23
When x is 1, y is 3.
01:26
When x is 2, y is 5.
01:28
When x is 3, y is 3.
01:30
And when x is 4, y is 1.
01:34
So using those estimates, we can calculate the trapezoidal rule estimate for the area under the graph.
01:42
T4 will abbreviate that.
01:45
Trapezoidal rule says we should take delta x over 2.
01:49
Then we add up those y coordinates in the following way.
01:51
We take f of x sub 0.
01:53
Then we multiply the ones in the middle by 2.
01:58
Come all the way across doing that until we hit the last one.
02:05
I'm determined to leave that out.
02:09
There we go.
02:10
We're not there yet.
02:11
There are one more 2 as a coefficient...