00:01
Let's start with the function f of x is equal to 5x squared.
00:06
Let's say we want to find the upper and lower sums.
00:11
So upper and lower sums for this on the interval 0 and 3.
00:21
This is a closed interval, by the way, with the square brackets.
00:25
So we're going to need a lot of information, but first we're going to write what the upper and lower sum is.
00:30
The lower sum, we'll start with that first, is going to be equal to the sum of the function applied at its minimum value times delta x.
00:44
And we're going to take i from 1 to n.
00:49
The upper sum is going to be pretty much the same formula.
00:52
We'll start at i equals 1 and we go to n.
00:54
But instead, this will be the function taken at the maximum value.
00:57
And we'll talk about what these are, delta x.
01:01
So this will give us our upper and our lower sum.
01:03
Now first we need delta x.
01:06
So delta x itself is the width.
01:09
Delta x is going to be equal to b minus a over n.
01:15
So b minus a, your b minus a, your b and your a value, let's actually move this over a bit so i can show.
01:24
Your b minus your a value is what we already have technically.
01:27
So we already have written our b and our a value.
01:35
So for us, this will be 3 minus 0 over n, which is just equal to 3 over n.
01:44
Now we'll leave it in this form for now in terms of n.
01:47
Let's go ahead and find our left endpoint, which is the minimum value we'll say.
01:54
So this is the left endpoint.
01:56
The left endpoint, which is little m sub i.
02:03
Little m sub i is going to be equal to 0 in this case, plus i minus 1 delta x.
02:14
So we don't have any initial part, initial condition.
02:17
And x equals 0.
02:19
We're just going to be equal to 0 as well.
02:22
But then we'll have that i minus 1 times delta x part.
02:27
That's the left endpoint.
02:28
The right endpoint, we don't have anything at the initial condition.
02:35
So right endpoint, big m, sub i will be equal to 0 plus just i delta x.
02:46
And so for both of these, we know the answer.
02:48
This will be i minus 1 times 3 over n.
02:54
And this one will be equal to i times 3 over n.
02:59
So we have our left endpoint.
03:01
Now we know what to put into f of x when we want to find this, these sums.
03:05
So let's go ahead and start with finding the lower sum.
03:09
The lower sum is going to be equal to the sum, i equals 1 to n of f of the left endpoint, delta x.
03:22
And so that's just going to give us the sum.
03:28
This is going to be f of, we'll write this as i minus 1 times 3 over n.
03:37
And delta x was 3 over n.
03:43
So since our function is 5x squared, we're just going to put the i minus 1 times 3 over n into that function.
03:55
And so this is going to be, instead of f, we'll have 5.
04:00
And we're going to have to square all of this.
04:03
So i minus 1 times 3 over n, all squared, and then 3 over n.
04:14
All right.
04:14
So this will give us, continue the sum.
04:19
Now we'll go ahead and square, because everything is multiplied, so it should be an easy square.
04:24
3 over n squared is 3 over n squared.
04:30
I minus 1, we'll leave alone for now, and then we have 3 over n.
04:38
So we can expand the i minus 1.
04:41
This will be the sum of, now we'll have 3 over n to the third power.
04:49
And i minus 1 expanded is i squared minus 2i plus 1...