- #1
Saketh
- 261
- 2
The Problem
Evaluate the surface integral of
[tex]
G(x, y, z) = \frac{1}{1 + 4(x^2+y^2)}
[/tex]
where [itex]z[/itex] is the paraboloid defined by
[tex]
z = x^2 + y^2
[/tex],
from [itex]z = 0[/itex] to [itex]z = 1[/itex].
My Work
I rewrote [itex]G(x, y, z)[/itex] as
[tex]\frac{1}{1+4z}[/tex].
Then, I evaluated the surface integral (I'm skipping a few steps in the evaluation here):
[tex]
\int \!\!\! \int_R \frac{1}{1+4z} \sqrt{1+4z} \,dA = \int \!\!\! \int_R \frac{1}{\sqrt{1+4z}}
[/tex].
My Confusion
I do not understand how to evaluate this integral properly. I am not experienced in multiple integration, but I have not found an issue with it until now.
Basically, what are my differential elements supposed to be ([itex]dx, dy[/itex]?). Am I supposed to use polar coordinates here?
If someone could put me on the correct track, I would appreciate it. Thanks!
Evaluate the surface integral of
[tex]
G(x, y, z) = \frac{1}{1 + 4(x^2+y^2)}
[/tex]
where [itex]z[/itex] is the paraboloid defined by
[tex]
z = x^2 + y^2
[/tex],
from [itex]z = 0[/itex] to [itex]z = 1[/itex].
My Work
I rewrote [itex]G(x, y, z)[/itex] as
[tex]\frac{1}{1+4z}[/tex].
Then, I evaluated the surface integral (I'm skipping a few steps in the evaluation here):
[tex]
\int \!\!\! \int_R \frac{1}{1+4z} \sqrt{1+4z} \,dA = \int \!\!\! \int_R \frac{1}{\sqrt{1+4z}}
[/tex].
My Confusion
I do not understand how to evaluate this integral properly. I am not experienced in multiple integration, but I have not found an issue with it until now.
Basically, what are my differential elements supposed to be ([itex]dx, dy[/itex]?). Am I supposed to use polar coordinates here?
If someone could put me on the correct track, I would appreciate it. Thanks!