00:03
So problem 5 .7, we're given a function here, f of x is 2 plus the sign of x.
00:10
So you know just by looking at that, if we're also looking on the integral from x is from 0 to pi, you should know that as a sign curve.
00:24
This just has been shifted up 2.
00:26
So if you look at that, it's going to start at 0.
00:30
Here's pi.
00:32
So the curve is going to start here, in here.
00:38
And then it's going to go up to three.
00:42
So look at this, this curve goes up, and then it comes down to one, and right there.
00:50
And sorry about that.
00:51
Back up, i did it, that was to two pi.
00:54
So if it goes to pi, sorry about that.
00:58
It's going to start at zero into zero, and it goes up value to three right here.
01:06
So what we're seeing is just one half of a cycle of the, curve.
01:12
Okay, so that's what it's going to look like.
01:13
They want us to figure out for different values of n, n is the number of rectangles that i break this up, what would be the upper sum and the lower sum.
01:20
So let's just take a look at this.
01:22
First of all, we want to look at when n is equal to two.
01:26
So if n is equal to two, i have two rectangles.
01:30
How do i get the upper sum and the lower sum? okay.
01:33
So what i'm looking at now is the right endpoint was used to determine the height of the rectangle.
01:39
So you can see the curve is.
01:41
Is increasing up to pi over two, decreasing from pi over two to pi.
01:47
So yeah, the right endpoint on that first rectangle gives me an upper sum.
01:52
In order to get an upper sum on the second rectangle, instead of using the right endpoint, i'm going to need to use the left endpoint to make that happen.
02:02
So it looks like i'm going to have to use pi over two in both cases as the height of the rectangle.
02:08
So in this case, if i do the estimate, let's go back.
02:11
To that and so if i look at that the estimate of the area is going to be the width of each rectangle so each rectangle is a width pi over two so the upper sum is going to be pi over two and then you evaluate the first height of the first rectangle is going to be f of pi over two and the same for the second to make it an upper sum so it's going to be f pi over two plus f of of pi over two.
02:44
And at pi over two, this is an easy one, the function evaluates to three, three plus three is six.
02:51
So you got six pi over two.
02:54
So this just turns out to be three pi.
02:58
Now to find the lower sum for that same case, to find the lower sum, again, this thing starts out increasing.
03:05
So what would happen if i switch to left sums? so to find the lower sum, you can see like, okay, at the height of that first rectangle has to be the left end point.
03:17
And the second rectangle has to, instead of being the left end point, it has to be the right end point.
03:24
So it's going to be f of zero and f of pi to get a lower sum.
03:29
So the lower sum is going to be same width...