calculus 3, double integral, Jacobian
so let's just go ahead and do a sketch first and because I don't know what color
so I'll just drag everything black first so this right here one over four x so this is the line
like this this is y equals one over four x next two X or slope is higher so it looks like this
y equals 2x now 4 over X it's a higher per hyperbola like this
and then one over X is the smaller one actually let me put on y equals 4 over X and then 1 over
X it will look like this look at this thing this is the region that we are talking about
this is the region that we're talking about how do we find it the area of this region just think
about in calc one so how to set the integrals if we do horizontal rectangles uh changes changes
vertical rectangles changes changes so we are totally stuck on this
don't worry there's something called the Jacobian that's going to help us and the
purpose of the Jacobian is to help us to change this crazy region to a much better
region so I have all this equation written in terms of why being isolated but if we don't
do that then you can see this right here it's the same as saying y over X is equal to 1 over
4 and then divide the X on process y over X is equal to two and then multiply the example sets
of X Y equals 4 and then lastly x y equals one guess what this this symbol this this similar
so what we are going to do is I'm going to call this to be let's just say U I will say that U
equal y over X yeah and then right here I will just call this right here to be v v equals x y
so this right here will be the change of variable that we have so now I have to pick this right here
all you have to do is we will have to get X and Y in terms of u and v so how do we do that just
solve system of equations multiply the X on both sides so we know this right here tells us y equals
x times U and now I will just put this right here for that so that will give us V equals x times Y
which is X U which is x square U yes V equals x square u x square equals V over U divided U
on both sides and then we see that X is equal to I'll just take the square root of V over square
root of U everything is positive so don't worry about the domain of that stuff don't worry yeah
so that's that and then let's just do so I'll put it here so this means y equals square root
of V over square root of U times U and this and that can be reduced so this right here tells us
y equals square root of V and u what we want to do next is we want a this integral to be in terms
of U and D after the transformation then to make that happen we have to get the circle Jacobian
okay this right here use the determinant notation now this time the bar the bar is not the absolute
voice just a determinate so we will do partial X partial U and then partial X partial V and then
partial y partial U partial y partial V and again this is the determinant notation to make this
easier to differentiate I'm going to write the following I will write this as a power notation
which is U to the negative one-half and then V to the positive one-half and then for the Y I already
as you going first possibly one half and then positive one-half and hope for the best here we go
partial x with respect to U so look at this as differentiate put the power to the front so we
have negative one over two and then minus one so we have U to the negative three over two and
these things just stays so like that now do this with respect to V so put the one half to the front
so we have one half this thing stays V and then minus one so minus that and then now do this with
respect to U we have one half U negative one half and this thing stays and now do this thing with
respect to V one half U to the one half and then V to the negative one half okay this times this
negative negative no negative positive so it's negative one over four yeah this V canceled
U to the negative three half times U to the one half is u to the negative one okay U to
the negative one and then we do this times that but it's a minus in between one half one half is
one over four this and that is U to the negative one and uh this and that and the cancel so we have
this and we can combine them negative one fourth minus one fourths is negative one-half and then
we have U to the negative one yes no we are not this is just a Jacobian it's just [Music] foreign
are we doing right yeah don't worry now integral integral still the function e but X Y hmm
it's like the V so we have negative V and then over two I'm going to just put on d u d v
original is like supposed to be like DX d y d y d x no the udv much better but we also
need this thing and remember this is just a Jacobian we need the absolute value and then
negative one half and we have U to the negative one
so that's the T by DX bar and to study up like this then U goes first so U starts with what
our U is this so it goes from one over four to one I mean to two so
I would just say U goes from one over four
to two and then V you see V is X Y so it goes from one to four so V goes from one
oh by the way now we've just made this positive and then U goes from this to that so it's past
the anyway so don't worry about the absolute value of U to a negative one this is this thing
has no U here so we can actually break them apart into two integrals so I'm going to do
this this one first so V is equal to 1 to 4 and now just to get e to the negative V over two
DV and then the other one I would just multiply it with U going from one over four
to two and uh I'll just look at absolute value of one half is just absolute value of negative
one of just one half and then we really don't need to worry about the absolute value for
because you know 1 over 42. you and then the good thing is if you integrate this
integrator multiplied you get the answer integrate this we have e to the negative
V over 2 but we will have to divide it by the derivative of this which is divided by negative
one over two which is same as multiplied by negative two and then we go from one to four
let's focus on this right here first so put four in here we
get negative 2 e that's negative two and then we
subtract but that's a negative so it becomes plus put a 1 here we get 2 e to the negative one-half
that's that this thing one half L and U and we go from
one over four that's radius 2 to the negative 2 and then 2 is just two
put two in here we get one half ln2 and then just minus one half I'll earn two to the negative two
but you see this negative 2 can go to the front so that's that and I can do the Blue part a little
bit more this is one half ln2 becomes plus and um that's gone and then we have just L and two
which is three over two ln2 so this is the blue part
and then the this part is still this this this this cancel and then let's
put a three back here to the power so ladies and gentlemen
we get Negative e to the negative two plus e to the negative one half times Ln eight