Volume of an egg

The previous post looked at an equation to fit the shape of an egg. In two dimensions we had

$\frac{x^2}{a^2} + \frac{y^2}{b^2}(1 + kx) = 1$

In this post, we’ll rotate that curve around the x-axis to find the volume. Then we’ll see how it compares to that of an ellipsoid.

If we rotate the graph of a function f(x) around the x-axis with x ranging from c to d, the volume is given by

$\pi \int_c^d \left(f(x) \right)^2 \, dx$

It works out well that the function f is squared because when we express y explicitly as a function of x we get a square root. Our volume works out to be

$b^2 \pi \int_{-a}^a \left(1 - \frac{x^2}{a^2} \right ) \frac{1}{1 + kx}\, dx = \frac{2b^2\pi \left(\left(a^2 k^2-1\right) \tanh ^{-1}(a k)+a k\right)}{a^2 k^3}$

We’d like to see how this compares to the volume of an ellipsoid, and that would be easier if we expanded the inverse hyperbolic tangent using its power series

$\tanh^{-1}x = x + \frac{x^3}{3} + \frac{x^5}{5} + \cdots$

This says our volume is given by

$2b^2\pi\left(\frac{2}{3}a + \frac{2}{15}a^3k^2 + \frac{2}{35} a^5 k^4 + \cdots \right )$

Note that if a = b = r and k = 0 this reduces to the volume of a sphere of radius r, i.e. 4πr³/3. If a and b are not necessarily equal but k = 0 we get 4πab²/3, the volume of an ellipse.

To first order, k does not effect the volume. That is, k does not appear in the series above except with exponent 2 and higher. This says to first approximation, the volume of an egg (assuming our formula for the shape) is simply that of an ellipsoid with the same major and minor axes. Also, k only appears to even powers. We should have expected that from the previous post since changing the sign of k just flips the egg over and doesn’t change the volume.

To second order, the volume of an egg relative to that of an ellipse is a quadratic function of the parameter k. To change an ellipse into an egg shape, making one end flatter and the other end more pointy, but keeping the length and width the same, you have to add volume. You gain more volume on the flatter end than you lose on the pointier end.

See the next post for a discussion of the surface area of an egg.