Comparison of differentiation & Integration (Part - 6) - Integrals, Maths, Class 12 Video Lecture

hello friends this video on integrals
part six is brought you by exam fear
calm no more fear from exams before
watching this video please make sure
that you have watched part one to part 5
let's compare definition and integration
so we know what is differentiation we
have learned this in depth we know
little about integration now so before
we proceed and try to solve some
integration numericals let's understand
or let's compare the differentiation and
integration the first thing is both our
operations on the functions what I mean
here is I can say f dash X that is
differentiation or again say integration
of FX DX it is integration so if you see
in both the case I am performing
operation on FX
correct second both satisfy the property
of linearity so if you see in case of
differentiation also if you have D by
D's of Cavin some function plus key to
some function you can break this you can
say K well into the derivative of first
function plus k2 into derivative of
second function similarly in case of
integration also if you have k1 some
function plus k2 some function and you
want to find integration of this or
integral of this there is nothing but
Cavin integral of first function plus k2
integral of second function that is both
satisfy the property of linearity third
is all functions are not differentiable
similarly all functions are not
integrated they are some functions which
are not continuous they are non
differentiable similarly they are
functions which are not integrated the
process of differentiation and
integration are inverse of each other
this is explained in the last few slides
that differentiation integrations are
exactly inverse of each other the
derivative of function when exists is a
unique function if you have a function
and if you want to find a derivative
that is a unique function y is the
integral of the function is not unique
but it is unique to
in there is it will have some constant
value but this part will be unique for
example and I am saying is if you have
FX is a good X cube right if X is going
to X cube you want to find derivative of
this you will get 3x square right
this is uni but if you have 3x square
and you want to find integration of this
you will get something like X cube plus
some constant using it X cube plus 1 X
plus 2 X plus C you can get lot of
functions which are since we have lot of
functions exactly not unique but unique
if you assume this butters constant so
there eret of constant actually so if
you know this path this is exactly you
think but if you take this part that you
are not unique but they are similar
third is when we talk of derivative we
talk of derivative at a point because we
have this function we have this graph
you want to find derivative of this we
want to find derivative of this stuff at
a particular point every point that you
may find different points if you take
the derivative will come out to be
different because derivative is nothing
but slope at that particular point so
when you talk up derivative at top of a
point then we talk of integrable
integral we don't talk of point we talk
of over an interval for example I have
this function I want to find integral I
talk about this particular interval so
there is a difference here when I talk
of derivative I talk of a point example
at this point they want to find slope
when I talk of integral I am talking
about an interval over which I want to
find integration geometrically if you
see derivative is nothing but slope of
tangent to the curve at a point
similarly if you see this integral this
is nothing but a family of curves
parallel to each other having a parallel
tangent at a point of intersection this
is what we have discussed earlier
so this thing we know that if I have my
curve I want to find derivative there is
nothing but slope at the point so
derivative has a point and at that point
you want find the slope of the tension
all right when I talk about you see
graphically here the integration I took
my example right into y is equal to 2x
and I got integral s X square plus C and
I got so many graphs like this right so
many graphs like this and the the common
thing here was the all where similar
graphs with some constant and you take
any line which is perpendicular to this
x-axis because X is variable axis and at
all this point the slopes where equal
that is how we geometrically represent
integration the derivatives is used for
finding a quantities like velocity of a
moving particle distance I will 6a Citra
where the integral is used to find this
since ever then velocity time is known
example if I have my distance time graph
I can use derivative to find D by D that
is velocity Y if I have velocity time
graph I I can use integration to find
distance error because this is
irrelevant V interview so when I am
multiplying these two I need integral
and I am dividing these two x and y if
you say Y by X is difference issue and Y
into X is integration that's how it is
derivation involves limits and
integration also involves limit so this
is kind of comparison between
integration and differentiation thank
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