## www.mathcentre.ac.uk/.../9.2%20Integration%20as%20the%20reverse%20of%20differentiation.mp4

• 0:02 - 0:07
By now you have had a lot
of opportunity to practice
• 0:07 - 0:08
differentiating common
functions.
• 0:09 - 0:12
And you have learned many
techniques of Differentiation.
• 0:13 - 0:17
So for example, if I were to
give you a function, let's call
• 0:17 - 0:18
this function capital F of X.
• 0:21 - 0:24
If it's a fairly simple
function, you'll know
• 0:24 - 0:25
how to differentiate it.
• 0:35 - 0:40
And so by differentiating it,
you'll be able to calculate its
• 0:40 - 0:44
derivative, which you'll
remember we denote as DF DX. So
• 0:44 - 0:48
that's a process that you're
• 0:49 - 0:55
Now in this unit we're going to
refer to DF DX as little F of X.
• 0:58 - 0:59
So little F of X.
• 1:00 - 1:04
Is the derivative of big
F of X. Let me write
• 1:04 - 1:06
that down little F of X.
• 1:08 - 1:10
Is the derivative.
• 1:18 - 1:19
As big banks.
• 1:21 - 1:25
And as I say, that's a process
that you're very familiar with.
• 1:25 - 1:29
What we want to do now is try to
work this process in reverse.
• 1:29 - 1:33
Work it backwards. In other
• 1:33 - 1:36
little F. And try and
come back this route.
• 1:39 - 1:43
And try and find the function or
functions capital F which when
• 1:43 - 1:46
they are differentiated will
give you little Earth. So we
• 1:46 - 1:51
want to carry out this process
in reverse. We can think of this
• 1:51 - 1:52
as anti differentiation.
• 1:59 - 2:03
So we can think of this anti
differentiation as
• 2:03 - 2:04
differentiation in reverse.
• 2:06 - 2:11
So big F of X we're going to
call the anti derivative of
• 2:11 - 2:12
little F of X.
• 2:27 - 2:31
So by these two results in mind
and try not to get confused with
• 2:31 - 2:34
a capital F in the lower case F
little F is the derivative of
• 2:34 - 2:40
Big F. Because little F
is DF DX.
• 2:42 - 2:45
And big F is the anti derivative
of little laugh.
• 2:47 - 2:51
Now you may have already looked
at a previous video called
• 2:51 - 2:52
integration as summation.
• 2:53 - 2:58
And in that video, the concept
of an integral is defined in
• 2:58 - 3:02
terms of the sum of lots of
rectangular areas under a curve.
• 3:03 - 3:07
To calculate an integral, you
need to find the limit of a son.
• 3:08 - 3:11
And that's a very cumbersome
and impractical process. What
• 3:11 - 3:15
we're going to learn about in
this video is how to find
• 3:15 - 3:18
integrals, not by finding the
limit of a sum, but instead
• 3:18 - 3:19
by using antiderivatives.
• 3:21 - 3:23
Let's start off
with the example.
• 3:24 - 3:28
Suppose we start off with the
function capital F of X.
• 3:30 - 3:34
Is equal to three
X squared plus 7X.
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Minus 2.
• 3:39 - 3:41
And what I'm going to do is
• 3:41 - 3:43
very familiar with. We're
going to differentiate it.
• 3:45 - 3:46
And if we differentiate it.
• 3:47 - 3:51
We get TF DX equals.
• 3:52 - 3:55
Turn by turn the derivative of
three X squared is going to be.
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2306 X.
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And the derivative of Seven X is
just going to be 7.
• 4:05 - 4:09
What about the minus two? Well,
you remember that the derivative
• 4:09 - 4:13
of a constant is 0, so when we
do this differentiation process,
• 4:13 - 4:15
the minus two disappears.
• 4:15 - 4:19
So our derivative DF DX is just
six X +7.
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So I'm going to write little F
of X is 6 X +7.
• 4:27 - 4:31
And think about what will happen
when we reverse the process when
• 4:31 - 4:33
we do anti differentiation.
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Ask yourself what is an anti
derivative of 6X plus 7?
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this question and anti
• 5:00 - 5:01
derivative of 6X plus 7.
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Is 3 X squared plus 7X minus
• 5:08 - 5:14
Is F of X is 3 X squared plus
7X minus two, but I'm afraid
• 5:14 - 5:18
that's not the whole story.
We've already seen that when we
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differentiate it, three X
squared plus 7X minus two, the
• 5:22 - 5:24
minus two disappeared.
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In other words, whatever number
had been in here, whether that
• 5:28 - 5:33
minus two had been minus 8 or
plus 10 or 0, whatever it would
• 5:33 - 5:36
still have disappeared. We've
lost some information during
• 5:36 - 5:39
this process of Differentiation,
and when we want to reverse it
• 5:39 - 5:44
six, XX have Seven, and working
• 5:44 - 5:47
backwards, we've really no idea
what that minus two might have
• 5:47 - 5:49
been. It could have been another
• 5:49 - 5:54
number. So this leads us to the
conclusion that once we found an
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antiderivative like this one,
three X squared plus, 7X minus
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onto this will still be an
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anti derivative.
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If F of X.
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Is an anti derivative.
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A little F of X.
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Then so too.
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Is F of X the anti derivative
we've found plus any constant at
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all we choose. See is what we
call an arbitrary constant. Any
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value of see any constant we can
add on to the anti derivative
• 6:38 - 6:41
we've found and then we've got
another antiderivative. So in
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fact there are lots and lots of
antiderivatives of 6X plus
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Seven, just as another example,
we could have had over here.
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Three X squared plus 7X minus 8.
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Or even just three X squared
plus 7X where the constant term
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was zero. So lots and lots of
different antiderivatives for a
• 7:05 - 7:07
single term over here.
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Let's do another example.
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Suppose this time we look at.
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A cubic function that suppose F
of X is 4X cubed.
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Minus Seven X squared.
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Plus 12X
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minus 4.
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Again, you know how to
differentiate this. You've
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differentiating functions
• 7:37 - 7:38
like this.
• 7:39 - 7:43
So the derivative DF
DX is going to be.
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Three 412 X squared.
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The derivative of minus
Seven X squared is going to
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be minus 14X.
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And the derivative
of 12 X is just 12.
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So think of our little F of X as
being the function 12 X squared
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minus 14X at 12.
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Notice again the point that
the minus four in the
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differentiation process
disappears.
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So now ask the question, what's
an anti derivative of this
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function here little F.
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Well, we've got one answer. It's
on the page already. It's this
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big F is an anti derivative of
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Little F. But we've seen
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here will still yield
another antiderivative, so
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we can write down lots of
other ones, for example.
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If we add on a constant.
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And let's suppose we add on the
constant 10 onto this one.
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Then we'll get the function 4X
cubed, minus Seven X squared.
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Plus 12X plus six is also an
anti derivative of Little F.
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And the same goes by adding
adding on any other constant at
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all we choose. If we add on
minus six to this.
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Will get 4X cubed, minus Seven X
squared plus 12X and adding on
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minus six or just leave us this
so that that is another
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antiderivative. There's an
infinite number of
• 9:26 - 9:27
antiderivatives of Little F.
• 9:29 - 9:30
Now in all the examples we've
• 9:30 - 9:34
looked at. I've in this in a
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right at the beginning because
we started with big F, we
• 9:37 - 9:41
differentiated it to find little
F, so we knew the answer all the
• 9:41 - 9:45
time. In practice, you won't
know the answer all the time, so
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one way that we can help
ourselves by referring to a
• 9:48 - 9:51
table of antiderivatives. Now a
table of antiderivatives will
• 9:51 - 9:52
look something like this one.
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There's a copy of this in the
notes accompanying the video and
• 9:57 - 10:00
a table of antiderivatives were
list lots of functions.
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F.
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And then the anti derivative.
• 10:04 - 10:08
Big F in the next column and
you'll see every anti
• 10:08 - 10:12
derivative will have a plus.
See attached to it where C is
• 10:12 - 10:15
an arbitrary constant. Any
constant you choose.
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What I want to do now is I
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of an antiderivative with
integration as summation.
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Let's think of a function.
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Y equals F of X.
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Let's suppose the graph of the
function looks like this.
• 10:51 - 10:54
I'm going to consider
functions which lie entirely
• 10:54 - 10:57
above the X axis, so I'm
going to restrict our
• 10:57 - 11:01
attention to the region above
the X axis were looking up
• 11:01 - 11:01
here.
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And also I want to restrict
attention to the right of the Y
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axis, so I'm looking to the
right of this line here.
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Let's ask ourselves what is the
area under the graph of Y equals
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F of X?
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Well, surely that depends on
how far I want to move to the
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right hand side. Let's
suppose I want to move.
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To the place where X
has an X Coordinate X.
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Then clearly, if X is a large
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number. The area under this
graph here will be large,
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whereas if X is a small
number, the area under the
• 11:42 - 11:45
graph will be quite small
indeed. Effects is actually 0,
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so this line is actually lying
on the Y axis than the area
• 11:49 - 11:51
under the graph will be 0.
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I'm going to do note the
area under the graph by a,
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so A is the area.
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Under
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why is F of X?
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And as I've just said, the area
will depend upon the value that
• 12:12 - 12:16
we choose for X. In other
words, A is a function of XA
• 12:16 - 12:21
depends upon X, so we write
that like this, a is a of X and
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that shows the dependence of
the area on the value we choose
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for X. As I said, LG X larger
area small acts smaller area.
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Ask yourself. What is the height
of this line here?
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Well, this point here.
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Lies. On the curve Y equals F of
• 12:43 - 12:48
X. So the Y value at that point
is simply the function evaluated
• 12:48 - 12:50
at this X value.
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So the Y value here is just F of
• 12:53 - 12:57
X. In other words, the
height of this line or the
• 12:57 - 13:01
length of this line is just
F of X. The function
• 13:01 - 13:02
evaluated at this point.
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Now I'd like you to think what
happens if we just increase X by
• 13:09 - 13:10
a very small amount.
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So I want to just move this
point a little bit further
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to the right.
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And see what happens.
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I'm going to increase X by a
little bit and that little bit
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that distance in here. I'm going
to call Delta X. Delta X stands
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for a small change in X or we
call it a small increment in X.
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By increasing axle little bit.
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What we're doing is we're adding
a little bit more to this area.
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contribution to the area. This
• 13:42 - 13:44
• 13:45 - 13:48
And that's a little bit extra
area, so I'm going to call that.
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Delta A, That's an incrementing
area changing area.
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I'm going to write down
an expression for Delta
• 13:58 - 13:59
a try and work it out.
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Let's try and see what it is,
but I can't get it exactly. But
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if I note that the height of
this line is F of X.
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And the width of this column in
here is Delta X. Then I can get
• 14:15 - 14:18
an approximate value for this
area by assuming that it's a
• 14:18 - 14:20
rectangular section. In other
words, I'm going to ignore this
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little bit at the top in there.
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If I assume it's a rectangular
section, then the area here this
• 14:27 - 14:28
• 14:29 - 14:33
Is F of X multiplied by Delta X.
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Now, that's only approximately
true, so this is really an
• 14:39 - 14:40
approximately equal to symbol.
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approximately equal to
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F of X Times Delta X.
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Let me now divide both sides by
Delta X. That's going to give me
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Delta a over Delta. X is
approximately equal to F of X.
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How can we make it more
accurate? Well, one way we can
• 15:04 - 15:07
make this more accurate is by
choosing this column to be even
• 15:07 - 15:10
thinner by letting Delta X be
smaller, because then this
• 15:10 - 15:13
I've got in here that I didn't
• 15:13 - 15:16
count is reducing its reducing
in size. So what I want to do is
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I want to let Delta X get even
smaller and smaller and in the
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end I want to take the limit.
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As Delta X tends to zero of
Delta over Delta X.
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And when I do that, this
approximation in here will
• 15:31 - 15:34
become exact and that will give
us F of X.
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Now, if you've studied the unit
on differentiation from first
• 15:41 - 15:45
principles, you realize that
this in here is the definition
• 15:45 - 15:50
of the derivative of A with
respect to X, which we write as
• 15:50 - 15:56
DADX. So we have the result that
DADX is a little F of X.
• 15:59 - 16:00
Let's explore this a
little bit further.
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We have the ADX is a little
F of X.
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What does this mean? Well,
it means that little laugh
• 16:14 - 16:16
is the derivative of A.
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But it also means if we think
back to the discussion that we
• 16:30 - 16:34
had at the beginning of this
video, that a must be the anti
• 16:34 - 16:38
derivative of F. A is
an anti derivative.
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F. In other words, the area is
F of X. The anti derivative of
• 16:53 - 16:56
little F was any constant.
• 16:59 - 17:03
That's going to be an important
result. What it's saying is that
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if we have a function, why is F
of X and we want to find the
• 17:08 - 17:11
area under the graph? What we do
is we calculate an anti
• 17:11 - 17:13
derivative of Little F which is
• 17:13 - 17:16
big F. And use this expression.
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To find the area.
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There's one thing we don't know
in this expression at the
• 17:21 - 17:24
moment, and it's this CC.
Remember, is an arbitrary
• 17:24 - 17:27
constant, but we can get a value
for. See if we just look back
• 17:27 - 17:29
again. So the graph I drew.
• 17:30 - 17:34
And ask yourself what will be
the area under this curve when X
• 17:34 - 17:38
is chosen to be zero? Well, if
you remember, we said that if
• 17:38 - 17:42
this vertical line here had been
on the Y axis.
• 17:42 - 17:44
Then the area under the
curve would have been 0.
• 17:46 - 17:49
So this gives us a
condition. It tells us that
• 17:49 - 17:51
when X is 0, the area is 0.
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When X is 0.
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Area is 0. What does this mean?
While the area being 0?
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X being 0.
• 18:10 - 18:14
Plus a constant and this
condition then gives us a value
• 18:14 - 18:18
for C, so C must be equal to
minus F of North.
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And that value for C and go back
in this result here. So we have
• 18:26 - 18:31
the final result that the area
under the graph is given by big
• 18:31 - 18:33
F of X minus big F of note.
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Let me just write that
down again.
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OK.
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Now let's look at this problem.
Supposing that I'm interested in
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finding the area under the graph
of Y is little F of X.
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Up to the point.
• 19:12 - 19:14
Where X has the value be.
• 19:15 - 19:20
So I want the area from the Y
axis, which is where we're
• 19:20 - 19:24
working from above the X axis up
to the point.
• 19:24 - 19:25
Where X equals B.
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We can use this boxed
formula here.
• 19:30 - 19:34
When X is be will get a of B.
That's the area up to be.
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Will be F of B.
• 19:41 - 19:44
Minus F of note.
• 19:45 - 19:49
So this expression will give you
the area up to be.
• 19:59 - 20:00
Suppose now I want.
• 20:00 - 20:04
Area up to A and let's
• 20:05 - 20:06
So now I'm interested.
• 20:08 - 20:13
In this area in here, which is
the area from the Y axis up to a
• 20:13 - 20:17
well, again using the same
formula, the area up to a which
• 20:17 - 20:19
is a of A.
• 20:20 - 20:24
Is F evaluated at the X value
which is a?
• 20:26 - 20:29
Subtract. Big F of note.
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So I have two expressions on
the page here, one for the
• 20:35 - 20:36
area up to be.
• 20:37 - 20:39
And one for the area up to A.
• 20:41 - 20:46
Ask yourself How do I find
this area in here? That's
• 20:46 - 20:48
the area between A&B.
• 20:49 - 20:54
Well, the area between A&B we
can think of as the area up to
• 20:54 - 20:58
be. Subtract the area up to A.
• 20:59 - 21:03
So if we find the difference of
these two quantities, that will
• 21:03 - 21:04
give us the area between A&B.
• 21:05 - 21:07
So we get the area.
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Under Wise F of X.
• 21:15 - 21:19
From X equals A to X equals
• 21:19 - 21:23
B. Is the area under
this graph from X is A
• 21:23 - 21:26
to X is B is given by?
• 21:27 - 21:30
Once the area up to be, subtract
the area up to A.
• 21:31 - 21:33
So if we just find the
difference of these two
• 21:33 - 21:36
quantities will have F of B
minus F of A.
• 21:41 - 21:45
Add minus F of not minus minus F
of notes. These terms will
• 21:45 - 21:49
cancel out. So In other words
that's the result we need. The
• 21:49 - 21:52
area under this graph between
A&B is just.
• 21:53 - 21:55
Big F of B minus big FA.
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Let me just write that
down again.
• 22:26 - 22:31
The area and Y equals F of X
between ex is an ex is B is
• 22:31 - 22:36
given by big F of B minus big F
of a where big F remember is an
• 22:36 - 22:37
anti derivative of Little F.
• 22:42 - 22:43
• 22:54 - 22:58
So this is a very important
result because it means that if
• 22:58 - 22:59
I give you a function little F
• 22:59 - 23:04
of X. And you can calculate an
anti derivative of its big F.
• 23:05 - 23:08
And you can find the area
under the graph merely by
• 23:08 - 23:11
evaluating that anti
derivative at B, evaluating
• 23:11 - 23:14
it A and finding the
difference of the two
• 23:14 - 23:14
quantities.
• 23:16 - 23:20
Now in the previous video on
integration by summation, the
• 23:20 - 23:23
area under under a graph was
found in a slightly different
• 23:23 - 23:25
way. What was done there?
• 23:25 - 23:31
Was that the area under the
graph between A&B was found by
• 23:31 - 23:35
dividing the area into lots of
thin rectangular strips?
• 23:36 - 23:40
Finding the area of each of the
• 23:40 - 23:45
up. And doing that we had this
result that the area under the
• 23:45 - 23:50
graph between A&B was given by
the limit as Delta X tends to
• 23:50 - 23:55
zero of the sum of all these
rectangular areas, which was F
• 23:55 - 24:01
of X Delta X between X is A and
X is be. If you're not familiar
• 24:01 - 24:06
with that, I would advise you go
back and have a look at that
• 24:06 - 24:09
other video on integration by
summation, but that formula.
• 24:10 - 24:13
For this area was derived
in that video.
• 24:14 - 24:18
And this formula
defines what we mean
• 24:18 - 24:21
by the definite
integral from A to B
• 24:21 - 24:24
of F of X DX.
• 24:25 - 24:28
That's how we defined
a definite integral.
• 24:29 - 24:32
And if we wanted to define a
definite integral, if we wanted
• 24:32 - 24:36
to calculate a definite integral
in that video, we had to do it
• 24:36 - 24:38
through the process of finding
the limit of a sum.
• 24:40 - 24:44
Now the the process of finding
the limit of a sum is quite
• 24:44 - 24:47
cumbersome and impractical. But
what we've learned now is that
• 24:47 - 24:51
we don't have to use the limit
of a sum to find the area. We
• 24:51 - 24:53
can use these anti derivatives.
• 24:54 - 24:57
Because we've got two
expressions for the area, we've
• 24:57 - 25:01
got this expression. As a
definite integral, and we've got
• 25:01 - 25:03
this expression in terms of
antiderivatives, and if we put
• 25:03 - 25:05
all that together will end up
• 25:05 - 25:09
with this result. The definite
integral from A to B.
• 25:10 - 25:12
F of X DX.
• 25:13 - 25:14
Is FB.
• 25:16 - 25:18
Minus F of A.
• 25:22 - 25:26
In other words, the definite
integral of little F between the
• 25:26 - 25:30
limits of A&B is found by
evaluating this expression,
• 25:30 - 25:34
where big F is an anti
derivative of Little F.
• 25:40 - 25:42
Let me write that formula
down again.
• 25:43 - 25:45
The integral from A to B.
• 25:47 - 25:54
F of X DX is given by
big FB minus big F of A.
• 25:56 - 25:57
Let me give you an example.
• 26:02 - 26:06
Suppose we're interested
in the problem of finding
• 26:06 - 26:11
the area under the graph
of Y equals X squared.
• 26:15 - 26:17
And let's suppose for the sake
of argument we want the area
• 26:17 - 26:22
under the graph. Between X is
not an ex is one, so we're
• 26:22 - 26:23
interested in this area.
• 26:26 - 26:30
In the previous unit on
integration as a summation,
• 26:30 - 26:34
this was done by dividing
this area into lots of thin
• 26:34 - 26:35
rectangular strips.
• 26:36 - 26:40
And finding the area of each of
those rectangles separately.
• 26:41 - 26:42
And then adding them all up.
• 26:43 - 26:48
That gave rise to this formula
that the area is the limit.
• 26:49 - 26:51
As Delta X tends to 0.
• 26:52 - 26:57
Of the sum from X equals not to
X equals 1.
• 26:58 - 27:00
Of X squared Delta X.
• 27:01 - 27:05
So the area was expressed as the
limit of a sum.
• 27:06 - 27:12
And in turn, that defines the
definite integral. X is not to
• 27:12 - 27:14
one of X squared DX.
• 27:16 - 27:20
Now open till now if you
wanted to work this area out
• 27:20 - 27:25
the way you would do it would
be by finding the limit of
• 27:25 - 27:27
this some, but that's
impractical and cumbersome.
• 27:27 - 27:30
this result using
• 27:30 - 27:30
antiderivatives.
• 27:32 - 27:35
Our little F of X in this
case is X squared.
• 27:40 - 27:45
And this formula at the top of
the page tells us that we can
• 27:45 - 27:47
evaluate this definite
integral by finding an
• 27:47 - 27:50
antiderivative capital F. So
we want to do that first.
• 27:50 - 27:53
Well, if little F is X
squared, big F of X, well we
• 27:53 - 27:57
want an anti derivative of X
squared and if you don't know
• 27:57 - 28:00
what one is, you refer back to
• 28:01 - 28:05
An anti derivative of X
squared is X cubed over 3
• 28:05 - 28:06
plus a constant.
• 28:14 - 28:19
So in order to calculate this
definite integral, what we need
• 28:19 - 28:20
to do is evaluate.
• 28:21 - 28:25
The Anti Derivative Capital F.
At B. That's the upper limit,
• 28:25 - 28:27
which in this case is one.
• 28:29 - 28:32
And then at the lower limit,
which in the in our case is
• 28:32 - 28:34
zero. Let's workout F of one.
• 28:36 - 28:38
Well, F of one is going to be 1
• 28:38 - 28:41
cubed. Over 3, which is
just a third.
• 28:42 - 28:43
Plus C.
• 28:45 - 28:47
Let's workout big F of 0.
• 28:48 - 28:50
Big F of 0.
• 28:51 - 28:54
Is going to be 0 cubed over
three, which is 0 plus. See, so
• 28:54 - 28:58
it's just see. And then we
want to find the difference.
• 28:59 - 29:01
Half of 1 minus F of note.
• 29:03 - 29:06
Will be 1/3 plus C minus C, so
the Seas will cancel and would
• 29:06 - 29:08
be just left with the third.
• 29:09 - 29:12
In other words, to calculate
this integral here.
• 29:13 - 29:16
All we have to do is find the
Anti Derivative Capital F.
• 29:17 - 29:21
Evaluate it at the upper limit
evaluated at the lower limit.
• 29:21 - 29:25
Find the difference and the
result is the third. That's the
• 29:25 - 29:28
area under this graph between
North and one.
• 29:30 - 29:32
Now let me just show you how we
would normally set this out.
• 29:33 - 29:35
We would normally set this
out like this.
• 29:39 - 29:44
We find. An anti derivative of X
squared, which we've seen is X
• 29:44 - 29:46
cubed over 3 plus C.
• 29:49 - 29:53
We don't normally write the Plus
C down, and the reason for that
• 29:53 - 29:55
is when we're finding definite
integrals, the sea will always
• 29:55 - 29:59
cancel out as we saw here, the
Seas cancelled out in here and
• 29:59 - 30:00
that will always be the case.
• 30:00 - 30:04
So we don't actually need to
write a plus C down when we
• 30:04 - 30:05
write down an anti
derivative of X squared.
• 30:07 - 30:10
It's conventional to write down
the anti derivative in square
• 30:10 - 30:15
brackets. And to transfer the
limits on the original integral,
• 30:15 - 30:18
the Norton one to the right hand
side over there like so.
• 30:20 - 30:24
What we then want to do is
evaluate the anti derivative at
• 30:24 - 30:28
the top limit that corresponded
to the FB or the F of one. So we
• 30:28 - 30:32
work this out at the top limit
which is 1 cubed over 3.
• 30:35 - 30:39
We work it out at the bottom
limit. That's the F of a or
• 30:39 - 30:40
the F of Norte.
• 30:41 - 30:45
We work this out of the lower
limits will just get zero and
• 30:45 - 30:47
then we want the difference
between the two.
• 30:48 - 30:52
Which is just going to give
us a third. So that's the
• 30:52 - 30:54
normal way we would set out a
definite integral.
• 31:00 - 31:01
So as we've seen.
• 31:03 - 31:07
Definite integration is very
closely associated with Anti
• 31:07 - 31:08
Differentiation.
• 31:10 - 31:14
Because of this, in general it's
useful to think of Anti
• 31:14 - 31:18
Differentiation as integration
and then we would often refer to
• 31:18 - 31:22
a table like this one of
antiderivatives as simply a
• 31:22 - 31:23
table of integrals.
• 31:24 - 31:27
And there will be a table of
integrals in the notes and in
• 31:27 - 31:30
later videos you'll be looking
in much more detail at how to
• 31:30 - 31:33
use a table of integrals. So
you think of the table of
• 31:33 - 31:35
antiderivatives and vice versa.
• 31:37 - 31:40
Very early on in the video, we
looked at this problem. We
• 31:40 - 31:44
started off with little F of X
being equal to four X cubed.
• 31:45 - 31:47
Minus Seven X squared.
• 31:48 - 31:51
Plus 12X minus 4.
• 31:52 - 31:54
We differentiate it.
• 31:57 - 32:02
To give 12 X squared
minus 14X plus 12.
• 32:03 - 32:08
And then we said that an anti
derivative of 12 X squared minus
• 32:08 - 32:09
14X plus 12.
• 32:10 - 32:13
capital F plus C?
• 32:14 - 32:17
We've got a notation we
can use now because we've
• 32:17 - 32:20
integrals and the notation
• 32:20 - 32:24
we would use is that the
integral of 12 X squared.
• 32:25 - 32:31
Minus 14X plus 12 DX
is equal to.
• 32:32 - 32:33
4X cubed
• 32:34 - 32:40
minus Seven X squared plus 12X
plus any arbitrary constant.
• 32:41 - 32:44
And we call this an
indefinite integral.
• 32:50 - 32:55
And we say that the indefinite
integral of 12 X squared minus
• 32:55 - 33:00
14X plus 12 with respect to X is
4X cubed minus Seven X squared
• 33:00 - 33:03
plus 12X. Plus a constant
of integration.
• 33:07 - 33:11
OK, In summary, what have
we found? What we found
• 33:11 - 33:12
that a definite integral?
• 33:14 - 33:20
The integral from A to B of
little F of X DX. We found that
• 33:20 - 33:22
this is a number.
• 33:23 - 33:28
And it's obtained from the
formula F of B minus FA,
• 33:28 - 33:33
where big F is any anti
derivative of Little F.
• 33:34 - 33:38
And we've also seen the
indefinite integral of F
• 33:38 - 33:40
of X DX.
• 33:41 - 33:46
Is a function big F of X plus an
arbitrary constant? Again, where
• 33:46 - 33:51
F is any anti derivative of
Little F&C is an arbitrary
• 33:51 - 33:56
constant. So in this video we've
learned how to do
• 33:56 - 33:59
differentiation in reverse.
• 33:59 - 34:00
antiderivatives. Definite
• 34:00 - 34:04
integrals. And indefinite
integrals in subsequent videos.
• 34:04 - 34:08
You learn a lot more teak
techniques of integration.
Title:
www.mathcentre.ac.uk/.../9.2%20Integration%20as%20the%20reverse%20of%20differentiation.mp4
Video Language:
English
Duration:
34:14
 mathcentre edited English subtitles for www.mathcentre.ac.uk/.../9.2%20Integration%20as%20the%20reverse%20of%20differentiation.mp4 EHCliffe edited English subtitles for www.mathcentre.ac.uk/.../9.2%20Integration%20as%20the%20reverse%20of%20differentiation.mp4 EHCliffe edited English subtitles for www.mathcentre.ac.uk/.../9.2%20Integration%20as%20the%20reverse%20of%20differentiation.mp4 EHCliffe edited English subtitles for www.mathcentre.ac.uk/.../9.2%20Integration%20as%20the%20reverse%20of%20differentiation.mp4 EHCliffe edited English subtitles for www.mathcentre.ac.uk/.../9.2%20Integration%20as%20the%20reverse%20of%20differentiation.mp4