I am trying to find the surface area of a 'biconcave disc', which is the shape of a red blood cell. I know the formula/length for the curve, which I am integrating to find the volume of the shape. To find the surface area of the shape, can I just multiply this length by $2 \pi$?
Following is the shape and curve I am talking about, with the first part showing a cross section of the 3D object, and the second showing the curve I am talking about.
I basically took the second curve and integrated it around the x-axis to find the volume.
You cannot. This follows already from the fact that it scales incorrectly – if you scale the whole thing up by $\lambda$, that quantity would scale with $\lambda$ whereas it should scale with $\lambda^2$.
Each curve element sweeps a surface area element of different radius. The correct way to take this into account is given in the Wikipedia article that achille hui linked to in a comment.