The first post in this series looked at a possible **formula** for the shape of an egg, **how to fit** the parameters of the formula, and the **curvature** of the shape at each end of the egg.

The second post looked at the **volume**. This post looks at the **surface area**.

If you rotate the graph of a function *f*(*x*) around the *x*-axis between *c* and *d*, the area of the resulting surface is

This integral cannot be computed in closed form for the function *f* describing an egg. (At least I can’t find a closed form, and neither can Mathematica.) But we can do it with numerical integration.

Here’s the Mathematica code.

f[x_, a_, b_, k_] := b Sqrt[(1 - x^2/a^2) / (1 + x k)]
area[a_, b_, k_] :=
2 Pi* NIntegrate[
f[x, a, b, k] Sqrt[1 + D[f[x, a, b, k], x]^2],
{x, -a, a}
]

As a sanity check, let’s verify that if our egg were spherical we would get back the area of that sphere.

`area[3, 3, 0]`

returns 113.097 and `N[36 Pi]`

also returns 113.097, so that’s a good sign.

Now let’s plot the surface area as a function of the parameter *k*.

Plot[area[4, 2, k], {k, -0.2, 0.2}]

The *y*-axis starts at 85.9, so the plot above exaggerates the effect of *k*. Here’s another plot with the *y*-axis starting at zero.

Plot[g[4, 2, k], {k, -0.2, 0.2}, PlotRange -> {0, 100}]

As with volume, the difference between an egg and an ellipsoid is approximately a quadratic function of the parameter *k*.