Surface element in cylindrical coordinates

[millifarads]

Homework Statement

$$\vec J_b = 3s \hat z$$
$$\int \vec J_b \, d\vec a$$

I need to solve this integral in cylindrical coordinates. It's the bound current of an infinite cylinder, with everything done in cylindrical coordinates and s is the radius of the cylinder. The answer should end up with a phi unit vector, but I just don't see how to get the actual unit vector.

Homework Equations

According to a pdf on MIT's site, the area element is:
$$d \vec a = s d\phi dz \hat s$$
There's a caveat, though, as the page says there's a choice of direction. Using this way makes the most sense to me, but if I use this, the dot product between the s and z unit vectors will cause the whole thing to go to zero.
http://web.mit.edu/8.02t/www/materials/modules/ReviewB.pdf

I've found another site that says to use the following, but I don't see the reasoning as to where the phi hat unit vector comes from.

$$d \vec a = s d\phi \hat z$$

The Attempt at a Solution

$$\int \vec J_b \, d\vec a = \int_0^s \int_0^{2\pi} s \hat z \cdot sdsd\phi \hat z = \frac{2 \pi}{3} s^3 \hat \phi$$

Using the first equation for the area element doesn't get me where I need to be, but the second one does.

Any help in figuring this out would be appreciated.

 

[andrewkirk]

Over what surface are you integrating? Your expression $\int \vec J_b \, d\vec a$ does not provide that crucial information.

In your first equation in 2, the integration variables are $\phi$ and $z$, which suggests the integral is on the surface of the cylinder.

But in the equation in 3, the integration variables are $\phi$ and $s$, which suggests the integral is a cross sectional disc bounded by the cylinder.