Undergrad Integration over a part of a spherical shell in Cartesian coordinates

[cromata]

I am modeling some dynamical system and I came across integral that I don't know how to solve. I need to integrate vector function f=-xj+yi (i and j are unit vectors of Cartesian coordinate system).
I need to integrate this function over a part of spherical shell of radius R. This part is actually projection of circle (that lies in x-y plane) of radius 'b' on this sphere. I know that it's usually easier to integrate spherical shell in spherical coordinates but it's really tricky to set the intervals of integration in this problem in spherical coordinates.
-So my question is, is it possible to integrate this in Cartesian coordinates. Because i know integration intervals: I just need to set the interval for x and y as circle in x-y plane (z interval would be little tricky)

I also posted image so you can better see what the problem is...

 

[Charles Link]

It can be integrated over $dx \, dy$ with $dA=\sec{\gamma} \, dx \, dy$, where $\gamma$ is the angle between the normal to the surface at any point and the z-axis. $\\$ It has been quite a number of years since I did such an integral in an Advanced Calculus course, but I still remember doing it this way. $\\$ Edit: I may have misread this: Are you doing a surface integral, or an integral over some volume? What I did is for a surface integral. If it is over a thickness of $\Delta r$ , you might be able to multiply the surface integral result by $\Delta r$ . $\\$ Additional edit: I believe in this case $\cos{\gamma}=\frac{z}{R}$.