Please do not solve... I know the solution and that after setting z=0 we get tha
ID: 3119708 • Letter: P
Question
Please do not solve...
I know the solution and that after setting z=0
we get that bounds of z are from 0 to 1-x-y
and that after setting y=0 the bounds of y are 0 to 1-x
and after setting x =0 the bounds of x are 0 to 1
and that we integrate in the order of dzdydx
BUT my question is why do we choose that order of integration? Couldn't we set y=0 first then have z and x depend on it?
and how come after we get z=1-x-y and we set y=0 we can use the previous equation and say y=1-x versus plugging it in the original and saying y=1-z-x. We would be using when both z AND y are equal to zero.. and same goes for x, the way he calculated it we would be saying that in order to get x boundary we have to look at where x y and z =0, why is this only true fro x? Why does the x boundary only come from all x y and z being zero but that dosn't work for th other two?
Don't solve just explain
dVExplanation / Answer
To decide the limitsof integration you are independent to decide the order of integration.
Just be careful about lower plane was given to be x=0, y=0 and z=0.
That means region bounded is between these three planes .
On the fourth side it had a plane which was x+y+z=1.
So starting with any limit first say if you start with searching for the limits of y , then lower bound is the plane y=0 and upper will be y=1-x-z.
Then after integrating it with repect to y , you will be left with only two variable there in the integral namely x and z.
Now again it is your own choice which one to integrate first .
Suppose you chose z to be integrated first at this point . Then obviously you have to convert the limits of z in terms of x as we are left with only one variable dx with respect we can integrate it again.
In such case the limits of integration will be varying from z=0(the lower bound given in question) and upper bound in terms of x as z=1-x.
Now you will be left with single variable in terms of x and you can integrate it easily and surely you will get the same solution.