Examples : Area bounded by a curve and a line Video Lecture - Mathematics (Maths) Class 12 - JEE

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sure that you have watched part 1 - Part
3 let's take one more example we have to
find here the area bounded by here X
square is going to fool you why why is
going to Y is equal to both and Y axis
in the first row so earlier I had y
square is equal to X form the heavy M
expressible Y function it's vodka this
is my X square is going to 4 by
professors then I have Y is going to use
my advisable to vote so I have 2 on this
name because this guy is a suppose 1 2 3
this guy mine is my Y is Oh in this line
is nothing but Y it's good I have drawn
this tonight now the question is is this
guy this guy and y axis Y axis is this
it is this day correct
see if you see what is the common in a
in this case the area has to be common
between this curve so let me draw that
with a blue pen let's do this my curve
that is common between this curve Y
square is will be fool you might in the
first part so this is my then if you see
Y is going to do so I will draw my is
going to do in the first card only so I
am drawing the first kind only by is
going to phone and turn the first color
like this and then my y-axis
so you see the area which is enclosed by
all these blue lines is nothing but you
see is this the one of the stamps
thus I can say this is the area because
this is the only area which has all
these things it's going to do X but
before this curve and this Y axis sorry
you can say that if we extend here Y if
we are not doing this guy because in
this guy I have Y as well too
Michael explains Mentawai echinus but
why is good to poor is not being in this
ball so the common area has to be the
curve which has all this required thing
this is my now in this case I will say
winery why am i marrying because they
might say X Mary my life is difficult
because if I exit area have to take this
area first and they say it because if my
ex is wearing I meant if I am wearing X
now if you see from here to here my
equation of curve is different here to
here the creation of curve is different
correct
let's understand I am saying X is
varying in that case my here to here the
curve is this way this curve this curve
from here to here my curve is different
that is why zerah do so that makes my
life difficult so what I'm doing is I'm
saying my why is that
because if I say Y very my curve is
always okay so my life is simple so let
us do this stupid with lucre only so
area is nothing but integration of X DUI
now the question is why is running from
what to what
so you see why is varying from 2 4 so
you see why I buy this wrong if X square
is got to 4 y X is nothing but root over
Y that is nothing but 2 so I will put
the value X Y DUI anyway to be a sponsor
to and take out so this becomes 2 into y
to the power 3 by 2 by 3 by 2 C 2 so
this becomes 4 by 3 into y to the body
right it is forward to God Q by 2 - this
is - Anthony - to the power 3 point C if
you saw there this is nothing worth 4 by
3 into this become to the power 3 that
is 8 minus this become to the par-3 8th
route that is true
that is nothing but if you see 32 - boy
to do is a 2x3
so it was not that difficult - so what I
had done I spent time in drawing the
curve and then I also spend time in
analyzing which I am looking for then I
400 I may be looking for this area the
area in the stars and then I found that
if I take if I make library my life is
system because there's only one curve if
I make if I wear it from Y from - the
whole but if I make X variable my X is
varying from it suppose from here to
here I see there are two curves one is
from here to there - from here today so
that in that case I have to find two
areas and add but I didn't want to do
that actually make Y about it let's take
home a example here you have to find the
idea of the region in the first quadrant
please note first quadrant enclosed by x
axis line X root 3 y and this circle so
I have this my XY axis and let's draw a
circle because this is nothing but
circle with 0 0 this one center and
small radius 1 2 3 4 almost that circle
I have made this now let's draw this
line X equal to root 3 by that is line
is passing from 0 to 0 because X Y is
the condition and has a slope and also
x-axis this guy so what is the area I'm
looking for so let us again draw in the
blue one first quadrant side intercept
only in Foursquare and x-axis let's draw
the x axis with the blue correct then my
line X root 3 why it's drawn with the
blue and then circle X squares below X
square plus y square beautifulness he
can draw the whole thing but I see the
intersection is only this area let's
draw this one
correct
even if I draw this it doesn't make
sense so this is my area I'm looking for
now if you see the trick part here is
you can if you very for example with
eggs or with wine but X with Y you can't
very actually why because you say with
respect to Y you don't have this curve
actually so with X equal only here to
bed but the trick part here is even you
very detects the function is not seen
because from here to here this point my
function is y function is this and from
here to here my Python services so a
function is also varying so I have to
say this is area 1 this is in here so
I'll say any required is nothing but
area 1 plus area correct this very area
1 was raised here and I have to find
both the areas so area 1 will be nothing
but if you see I'll just try it here
good y dot DX for this curve this line
is what Y is equal to root 3 y sorry y
is equal to 1 by root 3 X correct so
this becomes actually when the X and X
varies from 0 to this point 0 to this
point what is this one so we have to
first find that point also this point is
nothing but if you see this is this
point if you can find interest and this
point is only one intersection of this
line that is X is equal to root 3 Y and
this circle right this point you can
find the X current is same in these two
points so I can find this expert because
this is critical for me and what is the
value of this exponent 2 why I know this
is 2 and Y is 0 why because the length
is 2 radius is 2 X square plus y square
is equal to R square
this is too so to find this guy to find
this guy to find the intersection of
this circle and this line so let's first
find the intersection generally I can
put the value where right and if I had
this violet suppose this value is X we
shall find no or X dot that this guy's X
dot I'll find out of X dot now and then
a2 will be nothing but again y dot DX
but in this case my Y is different
y here will be if you see this is this
curve here for this curve Y is nothing
but du Oh minus X square correct this
way goes here for minus x value so this
guy will be of hole minus X square and
integrate this guy here it will be from
X naught to correct integration is
simple the only critical part here is
now finding the value of this point X 1
right this guy is X naught plus why not
let suppose there's also X naught and y
is 0 here actually this guy is X naught
conversable so let's mine X naught Y
naught so X naught Y naught is nothing
but point of intersection of these two
so let us do X square plus y square is
equal to 4 and X square is equal to 3 y
square I just squared this guy so point
of intersection of these two so let's
write it here X plus X square 0 3 y
square all right so let's put X square
is 3 y square n so this becomes 3 y
square plus y square is equal to 4 or I
get 4y square is equal to 4 or I get Y
is equal to 1 if Y is good to 1 my X
will be nothing but Y is equal to 1 X
will be nothing but X is equal to root 3
y that is root 3 into 1 so this guy is
root 3 so X naught is 0 this guy's root
XI comma 1 so I don't any Y naught so
leave it so I have this point now so my
area is nothing
even plus ay-two correct
that is this guy plus this guy so this
guy is why I went D X D X that is 1 by
root 3 X DX from 0 to X naught as I know
root 3 even plus a 2 that is root of 4
minus X square DX from 3 X naught 2 I
would just find the value of this this
is 1 by root 3 common outside because
it's constant this becomes X square by 2
from 0 to root 3 plus this becomes 4
minus X square integration becomes X by
2 root 4 minus X square plus 4 by 2 sine
inverse X by 2 and here this varies from
again mu 3 to 0 2 root 3 root 3 votes so
this becomes actually 1 by root 3 into
root 3 into 3 by 2 plus this guy is here
to solve 2 by 2 root 4 minus 4 plus 4 by
2 sine inverse 2 by 2 minus here is
deftly no 3 by 2 2 4 minus root 3 square
is 3 minus 4 by 2 sine inverse cube I
that is what you get so this becomes
root 3 by 2 actually correct Plus this
we have to find so this is what X 2 by 2
is 1 this is 0 anyway sine inverse 1 is
PI by 2 so this becomes 2 into PI by 2 2
into PI by 2 y 2 4 by 2 is 2 - Jessica
root 3 by 2 into 1 right
three-wide minus 4 by 2 is 2 sine
inverse root 3 by 2 is PI by 3 so if you
add everything what you get is PI by 3
what I see because root 3 by 2 minus
beta cancel and this is pi this is 2 pi
by 3 subtract to get PI by 3 so here
also what we observed was the difficult
part was trying this curve and finding
the that we have to find this particular
area once we know that need to find
these particular areas regard this area
in this area we divided this into area
even a - we found area of this area of 2
and then for this we have to find this
point where this is changing these
curves are changing if I want that at X
is 1 2 root 3 is changing so I integrate
this from 0 to root 3 this curve and
root 3 to do this curve we integrate yes
thank you
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