00:01
So here we'll go through the procedure of setting up a riemann sum, which approximates the area under a curve f of x between two finite limits.
00:27
So i have a function shown over there and two values on the x axis.
00:38
And we'd like to find the area under that graph.
00:44
So the first idea is that we're going to use rectangles.
00:49
So the rhyme and sum uses rectangular shapes.
00:58
So the area of each rectangle is equal to base times height.
01:10
So there are a couple things to decide about both the base and the height.
01:15
So the first decision that you have to make is how many rectangles.
01:20
So if somebody says use a rhyme on sum, you are going to have to do the decision -making.
01:32
If they tell you, use five rectangles, you're kind of stuck there.
01:37
So the delta x is the base of the rectangle.
01:42
I'll just show two rectangles for the sake of argument.
01:51
Each rectangle has the same base, and it's given by the difference between the two endpoints divided by n.
02:01
So b minus a tells you the full range of x and then you just divided into little pieces.
02:08
I believe that's called partitioning for obvious reasons.
02:14
And then for the height, the decision that you want to make is do you use the left endpoint or the right endpoint? and there are other choices such as the midpoint or to use both left and right.
02:38
And sometimes the shape of the function tells you how how to make that decision.
02:42
But i'll just show the left end point is you take your region on the left edge, each of those little regions, and you evaluate f of x left endpoint for each rectangle.
03:00
And then your rhyme and sum is the sum i equals one, to n delta x times f.
03:12
In this case, the left end points evaluated.
03:24
And i'll point out that the rhyme on sum is an approximation.
03:34
If you take the limit where the delta x approaches zero or n approaches infinity, that turns into an integral of f x dx from a to b.
03:54
So if you can actually get that integral in closed form, you can get an exact result.
04:02
So let's take a look at an example.
04:06
Let me pull it up because i like to use excel spreadsheets for this type of thing...