(x+y)^2 is not equal to x^2+y^2 you have to remember this because
otherwise the cat will be sad alright?
okay to expand X Plus y squared we will have to write this down twice so we have X Plus y times X Plus Y and then we are going to use the foil method
that means we are going to do the first times first x times x is x square and then the outer
x times y we are just going to write down plus x y and then the inner y times x is the same as x
times y so we'll just put on plus X Y again and last times last y times plus y squared and then
we see that this then that can be combined it so the answer is x squared plus 2 X Y plus y squared
okay to do X Plus y Square first we're going to say this much right here is X and this much right
here is y so the whole thing is X Plus Y and 2 square means that we are just going to get the
area of a square so right here let's put down X right here as well and then also white right
here as well and then now let's just make a big square like so aha really cool huh
and then we can see that this area is x times x so that's x square and this area is x times Y and
then this area is x times Y and then lastly this detail area is y times Y which is y squared so as
you can see all together we have x square and then we will have to add two of this so that's plus 2
x y and lastly we will have to add y squared and that's the answer all right in order for us to do
this we will first need some definitions so let me write them down right here let's define what
x is first and of course in calculus we'll try to Define this as an integral let's define X to
be the integral going from 0 to X of 1 DT next we need to have x to the second power and again we'll
Define this as an integral as well the integral going from 0 to X of 2T DT now we can only use
these two definitions and we see that here we have X Plus y Square so we will use this definition
this means we have this being the integral going from 0 to X Plus Y and then we have 2T DT
but here we can integrate and plug in because that defeats the purpose we can only use the
properties of integrals to take care of this how do well this integral goes from 0 to X Plus y
That's separate it let's talk about this as the integral going from 0 to X first and then 2T DT
and then we will have to add pick this up from X and then goes to X Plus Y and then we have 2T DT
this is perfect because this right here by definition is x square so that's great
but now what is this though as we can see this intake will start with X we don't want
that because in water for us to use these two definitions we will have to make sure the intake
will start with zero so right here let's just do some substitution and keep in mind we are in
the T world and we have ax right here so let's just go ahead and have T minus X so we can get
the zero and let's call this to be a new variable let's say U and now let's move this to the other
side so we get T equals U plus X differentiate this we will get DT equals du the derivative
of x in here will be 0 because X is like a constant here now we can take this and then
fix it and see what we get keep in mind though this is T is equal to X we will have to put it
here so that means U will be x minus X which is zero so we will have U goes to be 0 first
and then we will have to put X Plus 1 here the X will cancel so we'll just have the Y
and then we have the two and the t is U plus X and the DT is the same as the U very nice
now we're just going to take off this this way I'm going to distribute this so we have 2 U and
then that will be the integral going from 0 to Y and then 2u and let's close that we have the
view here and then we are going to add okay 2 times x but they are both constants in the
U world so you can take it on the outside right here and then look at the integral going from 0
to Y and then this right here is just going to be the U inside now what's this well this right here
is pretty much similar to this right which is something square and here we have the Y squared
then huh so this part gives us y squared and then the 2x is just 2X and now what's this part
well this right here is like we have a one so we are talking about this right here and when
we have the X here that will give us X but here we have the Y so this part will give us the Y
ladies and gentlemen this integral is precise d y square plus 2 X Y again we have the
correct answer so I'm pretty sure all of your t-shirts from pre-algebra to calculus do you
have told you that X Plus y squared is equal to x squared plus 2xy plus y squared but now
I wanted to tell you this right here might not be true because X and Y might not commute so
the best answer is X Plus y squared is equal to x squared plus X Y plus Y X Plus y squared
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