I have a function which depends on the solid angle $\frac{df}{d \Omega} = \frac{df}{d\phi d\theta} $. I want to integrate it over a part of the sphere limited by a rectangle. How should I set the limits of the integral given the position and size of the rectangle?
Lets say that the center of mass of the rectangle is at $(r = \text{dist}, \phi = 0, \theta = 90^ \circ)$ and it is $a$ by $b$ large.
My rectangle is perpendicular to the $XY$ plane, but isn't perpendicular to the $XZ$ not the $YZ$ plane.
Is the Cartesian system with a center in my rectangle a better system for the integral?
Edit: Here is a sketch of my countour. If you know any program which I should use to make a better drawing, I'll be happy to redo it. The corners of the rectangle touch the sphere, the rest of the rectangle is inside. I want to integrate over the surface confined by the graphical projection of the rectangle on the sphere.
Not Cartesian, the spherical coordinates are the natural and best choice.
It makes no sense to refer to large flat rectangles on a sphere.
If the width and breadth are in terms of spherical coordinate symbols we can refer to the limits as
$$ (\phi_1,\theta_1 ),(\phi_2,\theta_2) $$
and proceed to evaluate it. There is no need further to transform or render it into Cartesian coordinates.
EDIT1:
The limits are independent constants, whether in rectangular system or a spherical system.
However if you are referring to a great circle for instance which is not along parameter lines, $ \phi = f(\theta)$ or $ F(\phi,\theta)=0 $ should be known a priori.
Even in the plane if you wanted to evaluate the triangle area between lines $ x=-y,x=0,y=0,$ the contour relation should be known. It is no different here.