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Finding the area and center of mass of NACA 4 number symmetric airfoil
The equation for NACA 4 digit symmetric airfoils isref
1 20 ( − 0.103600000000000 r x c 4 + 0.284300000000000 r x c 3 − 0.351600000000000 r x c 2 − 0.126000000000000 r x c + 0.296900000000000 r x c ) N n c
where rxc is x/c or the ratio of position x to chord c and Nn is the NACA number or thickness ratio; by multiplying by two and integrating with respect to rxc going from 0 to 1 we obtain its area
0.00680883333333 N n c
and integrating the same equation from 0 to xcg and making it equal to the area divided by two we obtain the equation for the center of mass along the chord
1 10 ( − 0.02072 x c g 5 + 0.071075 x c g 4 − 0.1172 x c g 3 − 0.063 x c g 2 + 0.197933333333 x c g 1.5 ) N n c − 0.00340441666667 N n c = 0
from which Nn and c can be factored out
− 0.002072 x c g 5 + 0.0071075 x c g 4 − 0.01172 x c g 3 − 0.0063 x c g 2 + 0.0197933333333 x c g 1.5 − 0.00340441666667 = 0
and xcg can be found numerically to give 0.3994c for a closed trailing edge, as on the equation shown, and 0.4015c for an open trailing edge

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