Today was my test and I've absolutely no idea about if I'll pass it or not. (On the cut, or I don't know how to say)
I had to calculate the volume between the paraboloids and , if I remember well.
I've thought about it and I think that using Cartesian coordinates will be easier than spherical and cylindrical ones.
However I tried to calculate the volume via cylindrical coordinates: . I tried to set up the triple integral : goes from to .
from to .
And goes from to .... I wasn't sure. But I strongly believe I already made an error.
I have 2 questions : How can I calculate the volume they ask using cylindrical coordinates?
With what method would you solve the problem?
Thank you very much!
Oh, and before I forget, I know they cross each other with a circular ( ) section.
to determine the boundaries I prefer to sketch the graph like this
and I put in it the important points then I see z change from which curve to which curve in our graph we see that z change from
since the graph of the first curve is lower than the second one after that we do a dropping to the x-y plane since we first determine the boundaries of z if we drop of the whole solid that we have we will get a circle with radius 1 this is the intersect between two paraboloids like this
from the picture we see that y change from to
then x change from -1 to 1
in polar it is the same for z and for r and theta I think it is clear
I am sorry for my bad graph