How to calculate surface area of a curved plane? could anyone explain how to calculate the surface area of a curved plane?  I am trying to calculate the surface area of a "vaulted" ceiling that is 24' long, 7' wide, and the height of the curve is 4' at the mid-point so that I can figure out how much paint I need to buy.
Here is an illustration:  

If it were a simple flat ceiling, I would just multiply length x width, but that's not what I have here and a google search is leaving me empty-handed.  Any ideas?
Thank you very much!
-Neal

If I understand your description correctly, we can figure out the area as follows:
A rough estimate (taking the width as 8 instead of 7, which gives a half-circle) gives $\pi r l = 96 \pi \approx 300$.
$\tan \theta = \frac{4}{3.5} $, $r \sin \theta = \frac{\sqrt{4^2+(3.5)^2}}{2}$. Since $\sin \theta = \frac{4}{\sqrt{4^2+(3.5)^2}} $, we have $r = \frac{4^2+(3.5)^2}{8} = \frac{113}{32}$.
$\theta = \arctan \frac{4}{3.5} \approx 0.85917$.
Hence the length of the arc is $r (4 \theta) = (\frac{113}{32}) 4 \arctan \frac{4}{3.5} \approx 12.034$. Hence the required area is $\approx 24 \cdot 12.034 = 288.82$. This is consistent with the rough estimate.

