Okay, upon further thought of the problem, this can either be solved in 3D (sphere), or approximated in 2D (circle of radius r).
In 3D, I'm talking about a cut of distance h from the surface, towards the center (along a radius). So, imagine sawing off the skull cap of someone's head, and you're cutting a distance of h from the very top towards the center. I need the surface area of that.
In 2D, it's similar, except I'm looking for the area of a section of the circle, again a distance h from the border towards the center, along a radius. For this, imagine a sunrise, and the portion of the sun that has appeared over the horizon is a length of h from top of the arch to the plane of the horizon.
If there is any way to solve this surface area or area without integrating, that would be preferential, as I don't think this problem is supposed to make us get into integration. But any thoughts are much appreciated. Thanks!