Hi, I have to compute de volume for the region:
I've tried to do by two different parametrizations, in spherics and cylindrical coordinates
For cylindrical coordinates I've made:
For spherics I've considered
I've used wolfram to make the calculus, and it gives that for cylindrical coord. I'm having twice the volume that I have for spherical coord. I don't know where I'm committing the mistake. Bye, thanks for helping
Note: I'm multiplying by 3 because I was going to use for Gauss theorem and is the divergence for the vector field given by the problem
I think you're right Fernando. I haven't used the jacobian in any integral because I thought of it directly in cylindrical and spherical coordinates, I haven't made a change of variables at any step I think, should I involve the jacobian? I think I see it now, but I'm not completely sure, that I actually made a transformation of coordinates at first in both cases, while considering cartesian to polar and cylindrical coordinates. I think that the confusion drives from the fact that in the surface integral the jacobian matrix doesn't appear in the same way that it does on the ordinary integrals. Why is this? What a silly mistake I've made, thank you very much.
Alsosprach Zaratustra, I don't understand what you purposed
I will try to explain.
I say that instead of looking for volume of , which is a volume of dome of sphere, so we can to look at the following congruent problem in 2D instead in 3D:
First, draw yourself a sketch of a circle with radius , now draw a line (play the role of ).
Now, compute the volume of revolution with the functions: , and around y-axis.
See my sketch...
Well, I added the jacobians, but still giving different, I don't know why. The difference now is more sutil: In cylindrical coord. gives (clic here)
In spherical coord gives
I tried to proceed with cylindrical, but the exercise gets more complicated, so I want to be sure that what I'm doing with spherical is ok, so I can proceed that way, but still giving different answers.
I've also tried using a solid of revolution: http://www.wolframalpha.com/input/?i=Int_-4^4+Pi+%28%2816-x^2%29-1%29
I don't even know of any of this results is okey, perhaps they are all wrong :P