Modeling A Chicken Egg Math Ia Direct

This equation worked beautifully for the top half. The bottom half is symmetric (y negative). Once I had ( y = f(x) ) for the upper half (from ( x = -L/2 ) to ( x = +L/2 )), I used:

[ V = \pi \int_{-L/2}^{L/2} [f(x)]^2 dx ] modeling a chicken egg math ia

(around the x-axis):

[ y = \pm \frac{B}{2} \sqrt{\frac{L^{2} - 4x^{2}}{L^{2} + 8wx + 4w^{2}}} ] This equation worked beautifully for the top half

[ SA = 2\pi \int_{-L/2}^{L/2} f(x) \sqrt{1 + [f'(x)]^2} dx ] modeling a chicken egg math ia

IB Math AA/AI HL/SL