So let's look at this example.
We are using the trapezium
rule to approximate
and this is important to understand because
we're not accurately finding the area,
we're just approximating it.
The area between one and zero,
we've been given a function
and it says five ordinates.
So let's write this out.
This is what we're doing.
So we're looking for the
area between one and zero,
3x plus 1 to the power of 2 dx.
Now, all of this notation will make a
lot more sense when we do integration.
But basically, this sign here just
means find the area or the integral.
This is the integral sign.
These here are your limits.
So these are your X limits of zero and one
and I'll sketch that in
a second to show you.
This is obviously our function and this
thing here just says, "with respect to X."
So similar as the dx that we were using
in differentiation with respect to X.
We'll formally define this shortly
when we do integration the faster way.
So, first of all, the first fact that I
can use, they've given us five ordinates,
which to me means that we have
four strips or four trapeziums.
Now, I don’t really know
what this graph looks like
but I'm just going to roughly give us an
idea that we have some form of a function.
So, we are looking for the
area between zero and one,
so between there and there, for example,
and we're splitting it up into trapeziums.
Now, we've got four
So, if we make it better
so it goes through zero.
Let's make it a function
so it looks like that.
We've got four strips here.
So what we want to do is we want to
split this into four trapeziums.
That means we're going
to have five ordinates.
So we have zero, which is our first
ordinate, and one as our last ordinate.
So we have zero, one, two,
three, four, and five.
So we have one, two, three, four, and five.
We've got one extra ordinate.
Let's just fix that here.
So I should have five ordinates.
I'm just going to make sure
that we're doing that correct.
So we've got one, two,
three, four, and five.
Very thin strips but that's
what integration really is.
We're trying to make them
infinitesimally small here.
So, here, we have our
trapeziums or our strips
which means that we have five ordinates.
Now, we even told that this
goes from zero to one.
And the first thing that we need
to do is calculate the height.
So we want to know the height
of each trapezium, okay?
So, if we know what 0 to 1 is,
we need to know what the H is,
so what are these little
these little steps for
each individual trapezium.
So we want to know
what the height is.
Now, in order to do that, if
you really think about it,
how can you split this little length here
between 1 and 0 into these
number of trapeziums.
So, if I give you the formula for this,
you can say that the height
is B minus A over N.
B and A are your X limits.
So this is just something
that you can learn.
These are your X limits and
N is your number of strips.
So again, I'm sure you'd be able to
figure this out just by looking at it,
because you want to go between 0 and 1
and you want to split it
into four trapeziums.
So you can calculate the height from here.
We can say the height is, this is your
upper limit and this is your lower limit.
So you do one minus zero, B minus A, and
the number of strips we said was four.
So the height of each trapezium
is 1 over 4 or 0.25.
Let's call this step one.
So, when you're using the trapezium rule,
the first thing to do is calculate your
height because we need that for the formula.
Step two, we now have to
find all our Y values,
because let me just remind you of the area.
The area says of a trapezium
rule is approximately H over 2,
Y0 plus YN, that's the last one,
plus two times all the other, Y1,
Y2, however many that you have.
So that's just the
So, we've calculated our H.
We now need to calculate
all of these N values.
In order to calculate these N values,
you obviously need all your X values.
So, if we make a little table
and we start to say that we've got
all our X values and our Y values,
we're going to start
with our first X value.
So we want to know what the Y value is here
when we put in our first X value at zero.
And then we're going up by H
steps for each one of them.
So we're taking a small, little step
of H to go to the next X value.
So, if we go up in steps
of 0.25, I've got zero.
The next ordinate will be 0.25.
The next ordinate will be 0.5.
The next ordinate will be 0.75.
And the last ordinate will be 1,
which is what it says here
to go between 0 and 1.
And you can double check, you can count.
You've got one, two, three, four, five
ordinates, which is what they wanted.
To find your Y ordinates,
you now need to substitute these
values into your Y function.
So these here is your Y functions.
We're going to substitute
each one of them into Y.
So we're going to have three
3 times 0 plus one squared,
which in this case
just gives me one.
The next one, we have 3 times 0.25
plus one squared,
which will work out.
We've got 3 times 0.5
plus one squared
that will need to work out.
Once again, you can treat them as fractions
and it might be quite easy to add
them up, which we can do in a minute.
We've got 3 times 0.75
plus one squared.
And then we've got 3 times 1,
so brackets around this,
plus one squared.
I shouldn't have the plus in the inside.
Now, for a trapezium rule, you should
be allowed to use a calculator
when we're using one,
when we're using --
deriving the rule.
Up to here, we can do
this without a calculator
because we can just
work within fractions.
So, we can say that that's
just going to give us one.
This is the same as 3 times 1 over 4,
plus one all squared, which gives
you three over four plus one,
which is essentially just
7 over 4 all squared.
So we're just dealing with fractions here.
This is the same as 3 times a half
plus one, which is
3 over 2 plus 2,
which will be 5 over 2 squared.
The next one, if I write this
one out, so we've got three
and we've got 0.75
which is 3 over 4 ,
it's three quarters plus one all squared.
If you work this out,
that gives you 9 over 4,
which when you add to 1 will give
you 13 over 4 all squared.
And lastly, the last one should
be fairly straightforward.
We've got 3 times 1
plus 1 which gives you 4
squared here which is just 16.
So as you can tell that this will be a
little bit easier when we use a calculator,
but we can get away, we're not
doing it if we really have to.
But at no point, with an exam question,
I want you to use a trapezium rule
especially of this level of
complication without a calculator.
So, now that we know all our Y
values, this is my first Y value.
Y0, this will be Y1, Y2.
Obviously, I haven't worked
them out as numbers yet,
Y3 and Y4 are also my YN, if
you want to call it that.
Let's put it all into
the trapezium rule now.
So we can say that the area
is approximately H over two.
H is this value, so 0.25 over 2.
Take your first Y and your last Y.
So my first Y is 1, my last Y is 16 plus
two times all the other Y values here.
So, we're going to have 7 over 4 squared
plus 5 over 2 squared
plus 13 over 4 squared.
So all I need to do is work this out.
Do this in a calculator.
If I just rewrite
this as 0.25 over 2,
1 plus 16 gives me 17 plus 2.
I can square these.
Up to that point, I can
do -- that's 49 over 16
plus 25 over 4 plus 169 over 16.
And if you just put that all
into your calculator now,
we end up with a final answer
of approximately 7.0937.
And remember that if you were using a
calculator, a lot of it becomes easier.
So there are a lot of little calculations
that I've done that you wouldn't have to do
because you can just put the numbers
straight into the calculator
and find the area.
And you can see that it's
quite an exhaustive process.
If you have to do integration from scratch,
if you have to do integration
as an approximation,
I hope that you started to appreciate that
it's not the easiest of things to do,
and even if it's
easy, it's tedious.
It takes a long time, it takes a lot of
effort and you definitely need some help.
Or, if you're doing it from scratch,
it would just take much longer time.
So, what we'll do now is we'll move
on to a faster method of integration
which will make our
lives a lot easier
and hopefully you'll see us
coming to the same result
more accurately and more easily.