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another simpsons rule
In the picture attached, I have modeled a lake to have a total area of 50.35cm^2.
My intervals were 2.375cm, and the top area came out to 26.125cm^2, and the bottom to 24.225cm^2.
The scale used is 2.375cm = 10m
If I now have my total area in cm^2, how can I now convert this back to meters squared using this scale?

?? $\displaystyle cm \cdot \frac{m}{100\;cm}\;=\;m$
$\displaystyle cm^{2} \cdot \left[\frac{m}{100\;cm}\right]^{2}\;=\;m^{2}$
Do we dare suggest how one might convert such a thing if it were a "volume" problem?

I do not understand how to do this using my scale.
I know my area is 50.35cm^2, but I am having trouble converting this using the scale: 2.375cm = 10m.

The usual conversion for cm to m is as I have provided it, 100 cm = 1 m. The conversion from cm to m is as I have demonstrated it. This is just working in standard metric units.
Your problem is just working with a different unit conversion. You have 2.375 cm = 10 m. Just use this conversion in EXACTLY the same way as the standard conversion.
$\displaystyle
50.35\;cm^{2} \cdot \left[\frac{10\;m}{2.375\;cm}\right]^{2}\;=\;??
$
The important thing is to get the units in the right place, numerator or denominator.