What is the area of the portion of 1/8 of an sphere cut off by two parallel planes? So the problem that I'm trying to solve is as follows:
Assume 1/8 of a sphere with radius $r$ whose center is at the origin (for example the 1/8 which is in $R^{+}$). Now two parallel planes are intersecting with this portion where their distance is $h$ (Note:$h$ < $r$) and one of the planes passes through the origin.What is the area cut off by these two planes on the 1/8 of the sphere?
You may assume anything that you think is required to calculate this area, as given, my suggestion would be the angles at which the parallel planes cross xyz planes.
Obviously I'm not interested in trivial cases for example when the plane which passes through origin is one of xy, xz or zy planes.
I hope I explained everything clearly. I wish I could draw a picture for this but I don't how. Let me know if you need more clarification. Any hint or help about how to find this area is highly appreciated.

 A: I don't care to make the calculation for you, but I think(?) this is the region you intend:


Addendum.
Here is an idea to avoid computing a complex integral.
Let me assume that the slice "fits" in one octant, as I drew above
(as opposed, e.g., to exiting through the bottom of the octant).
First, imagine that you wanted, not the area in an octant, but the area in a halfspace,
demarcated by one of the coordinate planes.
Then, from a side view, we have this:

Your two planes are $p_1$ and $p_2$, with $p_1$ through the sphere center.
The halfspace boundary is coordinate plane $q$.
Now note that the difference between the half of the $2 \pi h$ area of the slice
and the truncated area is just the area of two spherical triangles (one illustrated,
one in back), whose angles
are all known.
So I believe the area in your octant differs from a quarter of $2 \pi h$ by two
spherical triangles, one at either end of the octant.  The dimensions of both triangles
are known from the orientation of the planes $p_1$ and $p_2$.
A: The portion of a sphere between two parallel planes is a spherical "zone". Restricting this portion to that inside one octant means that both planes are in the same hemisphere and the area you want is $1/4$ of the total area of the zone.
Is there a reason to use the techniques of multi-variable calculus to handle this problem? Mathworld's article on spherical zones gives the answer to your question ($2\pi rh$) and includes a derivation. Even though they call their variable of integration $z$, it seems to me like they're using the area of a surface of revolution technique that is taught in single-variable calculus.
