So the last question is asking us to use
the trapezium rule with five ordinates.
Now, we're using the trapezium
rule just in order to appreciate
how good integration actually is,
how much easier it makes our lives.
So we're going to practice
the trapezium rule again.
Remember that it is a
little bit tedious
and you will need to use
your calculator for this
because we're dealing
with square roots.
But we go through it step by
step and hopefully at the end,
we're fairly amazed with how the approximation
is very similar to the actual answer.
So let's try this question out.
We have the integral
between three and one,
one over root x.
And it says that we are
using five ordinates.
So remember what to do.
Firstly, just change your ordinates
to strips or the number of N
or trapeziums whatever
you want to call them.
So five ordinates means four strips
because the number of strips is
always one less than the ordinates
and this is also our N value.
Let's just go over the
formula for trapezium rule.
So the trapezium formula stated
that you have area H over 2,
Y0 plus YN plus 2 times
all the other Ys.
So however many you have, you
can just put them in there.
So that means that we need to find
everything in this formula step by step.
The first thing that we need
to do is calculate our h.
And if you look back
at our formula for h,
we said H is the height of each
trapezium, is b minus a over n.
B is the upper limit and
A is the lower limit.
So we'll do 3 minus 1 over
4 to give me 2 over 4,
which gives me an answer
of a half or a 0.5.
If you prefer to work in
decimals, you can use decimals.
If you like working in fractions,
then that's also fine.
We are going to put all these
numbers into the calculator
so it doesn’t matter
which version you use.
The next step, we need to
calculate all of these Y values.
Now, in order to find those Y values,
you firstly need to know all your Xs.
So, we're going to calculate each X
starting from 1 going up in steps of 0.5.
So, if we are to imagine what's
happening here on our graph.
So if we start with 1, our next ordinate
will be, when you add 0.5 to it,
so that will be 1.5.
Our next ordinate will be, when you add
another 0.5 to it, so that will be 2.
And these will be all your X
values until you get to 3.
And that's how you calculate
your X ordinates.
So if we put it in a little
table here to make it clearer,
we have our X values
here and our Y values.
Start with the
lowest value of X.
So, imagine we have some sort of a curve
and this is the area that
So start with the lowest
value of X which is one.
Our next value will be 1.5.
Next value will be 2.
Next value will be 2.5.
And the last value will be 3.
And I know that that's my last value
because that's as far as I need to go.
So as soon as you hit the
top value, you stop.
Check that it's the correct
number of ordinates.
So if it isn't, there's
something wrong with your H.
So go back and calculate your H again.
But if it is, then you've
done it correctly.
So five ordinates, one,
two, three, four, five.
So I'm confident that
my H values are correct
because I've got five ordinates
between 1 and 3.
The next thing I need to do is calculate
my Y, and my Y comes from here.
So my Y function is 1 over root X.
I should have a dx at the end.
So my Y value is
1 over root X.
So I'm going to put X here
to find each individual Y.
So the first one, I'll have 1
over root 1 to just give me 1.
The next one, I'll have
1 over root 1.5.
This gives me 0.8165 to 4 decimal
places when I put it in my calculator.
The next one will be
1 over root 2.
This gives me 0.7071 to 4 decimal places.
The next one will be 1 over root 2.5,
which according to me is
0.6325 to 4 decimal places.
And the last one is 1 over root 3,
which you can leave like this if
you want to stay more accurate
or because we've converted
everything into decimals.
That gives me 0.57735 or 34 --
I should do since I'm going
up to four decimal places.
Okay. All my Y values have been calculated.
Remember, this is going to
be Y0, Y1, Y2, Y3, and Y4.
So technically, this is my first
and this is my last value
and these are all the values in between.
So let's put it all into our formula
to just get rid of this graph here,
so we make some more space.
So when I put this into my formula,
I just need to replace all my Y
values with these numbers now.
So my area is approximately
H is 0.5 over 2.
Take your first Y value, which is
this one, and your last Y value.
So I have 1 plus 0.5774
plus two times all the
other Y values here.
So we're now going to add all
of them here in the bracket.
So 0.8165 plus,
so do that here,
0.7071 plus 0.6325.
Put it all into your calculator.
Just be careful that you do this step by
step so you don’t confuse your calculator.
This gives me an approximate area
of 1.472 to 3 decimal places.
So that's the area that I've
approximated, and you see, once again,
that it's a fairly
Now, if I do this quickly just using
our standard integration method.
So if we just do that here on the side.
If we do integrate one over root X dx
between the limits of 1 and 3,
the first thing just as differentiation,
I'm not happy with this
being at the bottom.
Remember, root X has a power to the half.
So I can bring that up.
So I can rewrite this as
X to the minus a half.
I haven't integrated.
I just brought it up, dx.
So remember what we've done root X is
the same as X to the power of a half.
If you brought it up from the
denominator, it will change sign.
So it was positive as a denominator,
when you bring it up, it becomes minus.
When you integrate, add 1 to
the power divide by a new power.
So minus a half, let's just write
that here, minus a half plus one,
that gives you positive a half,
so divided by a new power.
I don’t need to put C at the end
because this is a definite integral,
so I'm going to put numbers in.
When you have a fraction being divided,
a number being divided by a fraction,
this bottom number can
just come to the top.
So we can rewrite this as 2x to the half.
That half comes from here
minus a half plus 1.
And then I have to put
my limits in of 1 and 3.
So, this is the
same as 2 root X.
And I still haven't put my limits in,
just tidying it up as 1 of 3.
When you put your limits in, put three
in first then minus one from it --
not one, minus the
function with one in it.
So we can rewrite this as 2 root 3.
And then I'm going to take away 2 root 1.
So I've just replaced those Xs.
When you put this into your calculator, you
will see that you'll get answer of 1.464.
So, you can see how close they are really
up to one decimal place there
exactly the same 1.4 and 1.4.
They're a little bit different
from the second decimal place on,
but that makes me really believe
in my trapezium rule formula
that I've done it correctly.
Obviously, it's an approximation so
it's going to be a little bit out,
but you can always double check
with your general faster method
or you can check your faster
method if you have time
by using the trapezium rule and
any amount of ordinates really.
The more ordinates you have,
the more strips you have,
the more accurate your answer will be
because they'll be thinner and
you'll waste less, less area.
It will be more accurate.