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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's still so
many different methods that we could
study but we're going to start to lick
at basic integration because remember
what I said right at the start
calculus me is mainly built of two main
areas we have integral calculus and
differential calculus their big area so
there's so much to do in 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 so we're
almost imagining ourselves in the
ancient world and 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
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
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 you can go
back to the original function by
integrating
that's how closely linked they both are
so differentiation isn't 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 methods to approximate areas
under curves we have the Simpsons 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 will 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 axis 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 it 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 fit I
split it into trapezium so oh we go this
trapezium this trapezium distribution so
I've got here
for trapeziums if you've forgotten what
a trapezium is it's just this shape here
so it looks like that kind of this
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 ABCD and we find the area of
each individual shape because we know
the formula for the trapezium and then
we add it up together to approximate the
area remember this is 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 hundred percent 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 will see that although I
have four trapezium here so let's just
say we've got four strips or trapeziums
we now have 1 2 3 4 5 ordinates so we
have 4 trapeziums or 4 strips but these
x-values here we called ordinates so X
ordinates now this is an important
combination or thing to remember so the
number of trapeziums or the number of
strips is always 1 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 ordinance so if we call this
fact number 1 something that we need to
remember as we do calculations ok
for each X ordinate we are
also going to get an associate y-value
so let's just say this x 0 gives me y 0
so when I put X 0 into my function f of
X I end up with y 0 let's just be more
clear and call this function f of X when
I put X 1 into the function here I will
get a y-value there let's call this y 1
and you'll see that that happens for
each one of those Y values so Y 2
I have y 3 they're getting a little bit
closer Y 3 and then my last one here is
y 4 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 ordinate
now imagine that if split this trapezium
or this big shape into equal parts so
I've split it in two equal parts and
let's call each part each little height
of this a constant height so let's call
those H looks like M that's supposed to
be H and that's age 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 X 0
this is X 1 this value here gives you y1
and this value here gives you y 0 and
this gives you y 1 so I'm kind of trying
to zoom in into one of the trapezium so
this is trapezium a here ok 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 1/2 multiplied by the height
of the 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 any one
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 the area formula is going to
be 1/2 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 y 0 so let's just say y 0
and then the second height which we can
call B of this trapezium is y 1 and that
is the area of shape a let's do the same
for shape B we now have 1/2 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
three trapeze for trapeziums so we do
1/2 times H for the next one it will be
y2 plus y3 and for D we'll have 1/2
times h y3 plus y 4 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 1/2 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 all
together but you'll see that there's 1/2
times H in every one of them so I'm
going to write 1/2 times H here as a
common factor and then in brackets I'm
going to write all my Y so I've got y0
plus y1 plus y2 I'm here now plus that's
why 1 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 2 y 1
s 2 y 2 + 2 y threes so you can rewrite
this as 1/2 times H or H over 2 you now
have y 0 which is this value and then
you have one of them plus y 4 and then
you have 2 lots of Y 1 plus y 2 plus y 3
so I've kind of just factorize 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 that area equals and
this is the formula that we use remember
this is what we're using for for
trapeziums
so for strips ok but to apply this to
any number of strips you can treat this
as y 0 you can call this Y n 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 2 times all the other ordinates so
all the other all the other ordinates so
basically that's what the formula states
you have h over 2 brackets first Y plus
last y plus 2 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 M is always 1 less than the number
of ordinates you don't have to derive
this formula this is just for us to
where it comes from and but if you know
the formula all you then have to do is
substitute Y or Y or Internet into the
formula and you need to work out what H
is
well do your numerical example now to
explain how we actually apply this
formula
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