Surface area of a plane wing

Question

I want to determine the surface area of an aircraft wing using calculus, specifically focusing on the cross-sectional area (thickness distribution) formula. Although I am unsure if there exists a specific formula for this purpose, I am interested in understanding how to calculate the curved surface area of an airplane wing without relying on aerodynamic equations. I have provided images for those less familiar with aerodynamics and the aviation industry, which may aid in visualizing the three-dimensional aspects. In the figures, the cross-sectional area is defined using airfoil equations, incorporating a thickness distribution equation. Second image

It is noteworthy that the cross-sectional area of aircraft wings tends to increase as they approach the fuselage, with C1 and C2 representing the initial and final chord lengths. The variable L denotes the length of the aircraft wing.

General Thickness Distribution formula: $$y_{t}=\pm5ct[0.2969\sqrt{α}-0.1260α-0.3516α^{2}+0.2843α^{3}-0.1015α^{4}]$$

Thickness Distribution formula for NACA 2412 (which I want to find): $$y_{t}=\pm0.60[0.2969\sqrt{α}-0.1260α-0.3516α^{2}+0.2843α^{3}-0.1015α^{4}]$$

Where c is the chord length, α = x/c and t is the maximum thickness expressed as a fraction of the chord length. In other words, 1/100 last two digits×chord length.

It is not necessary to use the given values, but you can if you want.

$$L = 30m$$ $$C1= 3m$$ $$C2= 1.5m$$

Answer 1

Considering $f\left(x,c\right)$ where $c = c_2+\frac{(c_1-c_2)l}{L}$ as the upper wing profile parametrized by the position along the wing span, the length element is given by

$$ \sigma = \sqrt{\left(\frac{df}{dx}\right)^2+1}=\sigma(x,c) $$

then the upper wing area is given by

$$ \int_{l=0}^{l=L}dl\int_{x=0}^{x=c_2+\frac{(c_1-c_2)l}{L}}dx \sigma(x,l) $$

Follows a MATHEMATICA script which calculates a good area approximation. Here we consider the upper wing profile.

f[x_, c_] := 0.6 (0.2969 Sqrt[x/c] - 0.1260 x/c - 0.3516 (x/c)^2 + 0.2843 (x/c)^3 - 0.1015 (x/c)^4)

sigma[x_, c_] := Sqrt[1 + 0.36` (-(0.126`/c) - (0.7032` x)/c^2 + (0.8529` x^2)/c^3 - (0.406` x^3)/c^4 + 0.14845`/(c Sqrt[x/c]))^2]

c2 = 1.5;
c1 = 3;
L  = 30;
Quiet@NIntegrate[NIntegrate[sigma[x, c2 + ((c1 - c2) l)/L], {x, 0, c2 + ((c1 - c2) l)/L}], {l, 0, L}]

(* 67.8476 *)

NOTE

A coarse area approximation is obtained by doing

(c1 + c1)/2 L // N  

(* 67.5 *)