00:01
To approximate the definite integral from 4 to 16 of 1 over x dx, using remand sum with 4 subintervals of equal length, the first step will be to get the length of these subintervals, that's defined by delta x, which is equal to b minus a over n.
00:21
Now this is just 16 minus 4 over n which is 4, that's equal to 12 over 4 or 3.
00:30
So from here we can determine our endpoints of the sub -intervals.
00:38
These are x -0 which is equal to the left endpoint 4.
00:45
And then our x sub 1 will be a distance of 3 from 4, so it will be 4 plus 3, which is 7.
00:54
X sub 2 will be a distance of 3 from 7, so it will be 7 plus 3 equal to 10.
01:00
X sub 3 will be a distance of 3 from 10.
01:04
It will be 10 plus 3, which is 13.
01:07
And x sub 4 is a distance of 3 from 13, or the right endpoint, which is 16.
01:17
So if you are to illustrate this, we have this line segment from 4 to 16.
01:24
And so the other endpoints will be in between them.
01:28
We have here x sub 1, x sub 2, and then x sub 3, which are 7, 10, and 13 respectively.
01:41
And for the midpoints, we need to take the sum of the two endpoints of the subinterval, and then we divide it by 2.
01:49
That will be the middle number.
01:51
So for this one, you do 7 plus 4, which is 11, and then you divide by 2, you get 11 over 2...