So after all that differentiation,
it's now time to start giving our attention to integration.
We have done a lot of differentiation,
and there are still so many different methods that we could study.
But we're going to start to look at basic integration,
because remember what I said right at the start,
calculus is mainly built of two main areas.
We have integral calculus and differential calculus.
They're big areas, so there is so much to do
and so many different methods and functions and kind of methods to learn.
But if we start with basic integration,
we'll build up just the way we did with differentiation.
Remember, we're going to discuss or follow the same sort of pattern
that we did with differentiation.
We'll look at why we integrate.
Then we'll look at the methods of integration from the basics,
and then we'll move forward to faster, easier methods of integration.
So let's start.
We're going to look at the very basic principles of integration.
Now, integration essentially means finding the areas under or above curves,
but we'll talk about where these areas are when I sketch it out for you.
So the first thing we're going to do is look at the trapezium rule
and then we'll build up to the general rule.
Trapezium rule is an approximation of the area.
See, we're almost imagining ourselves in the ancient world.
The kind of questions that perhaps the
ancient Greeks or the ancient Egyptians may have asked
as to how you calculate the area of a more complicated shape.
Squares, circles, and triangles, we can do.
But how do you calculate areas underneath curves or above curves?
The technique that we'll be using is obviously going to be integration,
and then the application of it is finding areas
under more complicated curves or functions.
So, a little bit about integration before we start.
Mathematically, it is the opposite of differentiation.
So, with differentiation, when we brought the power down and decreased it by one,
we'll see how we can do the opposite to find out the integrals.
As I've mentioned before, integration is going to find the area
underneath or above curves,
and I'll explain that a little bit better when we can actually see a graph.
Integration can also be used to solve differential equations.
So because I said it's the opposite of differentiation,
once you've differentiated it,
you can go back to the original function by integrating.
That's how closely linked they both are.
So, differentiation is an irreversible process.
If you've differentiated something,
you can go back to the original function by integrating.
So let's start with the first type of approximation.
This is perhaps the easiest type of approximation
and it's known as the "trapezium rule".
There is a lot of different method to approximate areas under curves.
We have the Simpson's rule.
We have lots of different types of more advanced methods
to find more accurate areas under curves.
But if we start with the derivation of this,
we'll, hopefully, get an idea of more advanced methods as we go along.
Imagine we have a curve.
So if I just draw the X and Y axes.
And imagine that we have some form of a curve here.
Now, the question you may ask,
"How do you find or how do you approximate the area under the curve?"
Now, if you think about it, and I imagine that's
what the people in the ancient world did,
they must have come up with some idea of splitting this curve into little shapes,
little shapes that they can deal with.
For example, trapeziums.
So what if I split this curve and I say, "I want to find the area from here to there"?
But to make it more accurate, I said I split it into trapeziums.
So, we go with this trapezium,
this trapezium,
and this trapezium.
So I've got here four trapeziums.
If you've forgotten what a trapezium is, it's just this shape here.
So it looks like that kind of those trapezium stood up.
Usually, we're used to seeing it that way,
but I've got them standing up.
So this here is a trapezium.
So, as you can see under my curve, I split it into four trapeziums.
And the idea is that we find the area of each shape.
So let's just call it A, B, C, D.
And we find the area of each of individual shape
because we know the formula for the trapezium,
and then we add it up together to approximate the area.
Remember, this is an approximation
because there are little parts here that are either sticking out or underneath,
which means that it won't be a 100% accurate
but it will give us some form of an approximation of the area.
So, how do we do this?
We need some values for each trapezium.
Let's start to put in some X and Y coordinates here.
So, if I say, this value here is X0
and my next value is X1, X2,
so they're all X coordinates, X3 and X4,
you'll see that although I have four trapezium here.
So let's just say we've got four strips or trapeziums.
We now have
one, two, three, four, five ordinates.
So we have four trapeziums of four strips.
But these X values here we called ordinates.
So X ordinates.
Now, this is an important combination or a thing to remember.
So, the number of trapeziums or the number of strips
is always one less than the ordinates.
So if I just write that here as a little note.
So we have number of strips.
Let's just call them --
is 1 less than the ordinates.
So, if we call this fact number one,
something that we need to remember as we do calculations.
Okay.
For each X ordinate, we are also going to get an associate Y value.
So let's just say this X0, gives me Y0.
So when I put X0 into my function f of x I end up with Y0.
Let's just be more clear and call this function f of x.
When I put X1 into the function here, I will get a Y value there.
Let's call this Y1.
And you'll see that that happens for each one of those Y values.
So Y2, I have Y3.
They're getting a little bit closer.
Y3.
And then my last one here is Y4.
So you sort of get the idea that where for each value we're getting --
for each X value, we're getting an associated Y value.
So that's that point here.
Okay.
Now, we can look at each trapezium.
We know that for each trapezium we have the X ordinates
and we have the associated Y ordinates.
Now, we imagine that I've split this trapezium or this big shape into equal parts.
So I've split it into equal parts,
and let's call each part, each little height of this a constant height.
So let's call those H.
It looks like M
It's supposed to be H.
And that's H there.
Okay.
Let me just zoom in on one of the trapeziums.
So, we're saying that this here is H.
Let's just look at the first one.
This is X0.
This is X1.
This value here gives you Y1.
And this value here gives you Y0 and this gives you Y1.
So, I'm kind of trying to zoom in into one of the trapeziums.
So this is trapezium A here.
Okay.
Now that I've got everything here,
we can actually start to derive our trapezium rule.
So, what people may have thought of before is that we
find the area of each individual shape and then we add it up together.
So if I take the area of A.
Now, a trapezium area, so if we just define what that is,
is a half multiplied by the height of trapezium which is this height,
or this height if it's standing up,
A plus B.
And A and B are these values here.
So one of them is A and the other one is B.
So, in our case, we've got an A value anyone of them,
so we can call this value little A and we'll call this value little B.
So remember, all we've done is we've stood a trapezium up on its height.
Okay.
So let's start to find the area of A.
The area of A following area formula is going to be half
times the height which we are also calling H.
And then you've got your A and B.
So A value here is the height of this trapezium.
So the height of this trapezium is Y0.
So let's just say Y0.
And then the second height, which we can call B of this trapezium, is Y1.
And that is the area of shape A.
Let's do the same for shape B.
We now have a half multiplied by H.
And in this case, your height or your A value of B will be Y1.
And then the second one will be Y2.
And obviously, we do that for all of them.
So we've got four trapeziums.
So we do half times H.
For the next one, it will be Y2 plus Y3.
And for the D, we'll have a half times H,
Y3 plus Y4.
And then the idea is that we add all of these areas together
to get an approximation for the full area.
So, if you start to do that, we're going to add all of them up.
Now remember, there's a half times H in all of them.
So I could just take that as a common factor.
So I can say that the area is approximately.
You can write it altogether,
but you'll see that there's a half times H in every one of them.
So I'm going to write a half times H here as a common factor.
And then in brackets, I'm going to write all my Y's.
So I've got Y0 plus Y1, plus Y2.
I'm here now.
Plus, that's Y1 again,
plus Y2, plus this one,
another Y2, plus this term, Y3,
plus this term, Y3, plus another Y4.
Okay.
You'll see now that you have two Y1s, two Y2s, and two Y3s.
So, you can rewrite this is a half times H or H over 2.
You now have Y0, which is this value.
And then you have one of them, plus Y4.
And then you have two lots of Y1, plus Y2, plus Y3.
So I've kind of just factorized them.
And this is our first formula.
This is the formula for the trapezium rule.
It's an approximation of area.
So we use that rather than saying the area equals.
And this is the formula that we use.
Remember this is what we're using for four trapeziums, so four strips.
Okay?
But to apply this to any number of strips, you can treat this as Y0.
We can call this Yn, so that's the last value.
So essentially, you're doing H over 2, first value,
plus the last value,
or the last Y ordinates, and then it's two times all the other ordinates,
so all the other ordinates.
So, basically, that's what the formula states.
You have H over 2 brackets first Y plus last Y
plus two times all the other ordinates.
In our case, obviously, we have four strips and five ordinates.
So, this is what it looks like.
We've put numbers in.
But in any other case,
you can always just work them out
by just substituting the numbers in.
So, we'll move straight into an example.
But before that, I just want to recap what everything is in the trapezium rule.
Remember that you have the number of strips and the number of ordinates
relationship which is quite important.
So depending on what the question says, if they give you strips or trapeziums.
I'm just calling them strips here,
is always one less than the number of ordinates.
You don’t have to derive this formula.
This is just for us to see where it comes from.
But if you know the formula,
all you're going to have to do is substitute your Y ordinates into the formula
and you need to work out what H is.
We'll do a numerical example now to explain how we actually apply this formula.