I have run into an interesting volume problem that I can't seem to figure out. I was recently in a school gym and noticed the interesting design of the ceiling and thought back to Calculus III and finding the volume under surfaces.
I have been trying to come up with a rough equation that models the design of the ceiling but have not been able to. The floor is roughly 100ftx100ft and the center of the ceiling is roughly 28ft. I believe that the four curved sections are parabolic. The equations I have been playing with are $ f(x,y) = (-1/16)x^2 + 6$ and $ f(x,y) = (-1/16) y^2 + 6$.These have just been jumping-off points and I figured their intersection region is roughly the shape of the ceiling. I'm not sure if trying to parameterize these equations is the way to go, or if I'm totally off base altogether. Any help would be appreciated!
Assuming cross sections of the ceiling are indeed parabolic, suppose the wall adorned with the cross is the north wall. In the $x,y$-plane, this wall corresponds to the line $y=50$. The intersecting arches correspond to the lines $y=\pm x$. So it looks as though the north quadrant of the ceiling could then be modeled by $f(x,y) = 28 - kx^2$ (a parabolic cylinder) over the region $y\ge|x|$, and the volume under it would be
$$\int_{-50}^{50} \int_{|x|}^{50} f(x,y)\,dy\,dx$$
By symmetry, the total volume is $4$ times this integral. You can determine $k$ from $f(\pm50,0)=0$.
The function modeling the ceiling, reflected into each quadrant, looks something like this: