00:01
Where as to write down the riemann sum using the right end points and take the limit and calculate the area under the curve.
00:09
First of all, we'll make a picture.
00:12
Here is the interval from 0 to 5 and we'll split it into n subintervals.
00:22
So there are n subintervals and the length, therefore, the length of each subinterval is the length of the whole interval, which is 5, divided by the number of subintervals, which is n.
00:34
That's going to be our delta x.
00:38
Now let's talk about the right end points.
00:41
The length of each subinterval is delta x, which is 5 over n.
00:49
Therefore, the first end point, since we are using the right end points, will be 5 over n, which means x0 equals 5 over n.
01:03
Whereas it's better to start with x1.
01:06
X1 equals 5 over n.
01:08
Then x2 will be equal to multiplied by 5 over n, because we need to make, to cross two subintervals.
01:17
So this point here is 2 times 5 over n.
01:24
This point here is 3 times 5 over n.
01:28
Note that we are using right subintervals, right end point of each subinterval and so on.
01:35
The last one, therefore, will be 5 and 5 is precisely n multiplied by 5 over n.
01:43
So that's the nth subinterval, the nth right end point.
01:48
Then the area, the riemann sum equals the limit of the sum i from 1 to n, n goes to infinity of f of xi multiplied by delta x.
02:14
This equals the limit as n goes to infinity of the sum i from 1 to n, f of xi.
02:24
To get f of xi, we need to plug in, so let's write down xi equals i multiplied by 5 over n.
02:33
And i here starts with 1 and goes all the way up to n.
02:39
Recall this subinterval, the right end points.
02:44
So we will plug in that xi, i times 5 over n for x in the function f...