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This question comes up from time to time. It seems quite straightforward. To compute the surface area of a body of revolution, you must integrate the function

$$S=2\pi \int r\sqrt{1+{\left(\frac{\mathrm{dr}}{\mathrm{dx}}\right)}^{2}}\mathrm{dx}$$from nose to tail.

For a Sears-Haack body (fixed length and volume)

$$r={[4x(1-x)]}^{\frac{3}{4}}$$ $$\frac{\mathrm{dr}}{\mathrm{dx}}=3{[4x(1-x)]}^{-\frac{1}{4}}(1-2x)$$where r is radius / max.radius and x is the distance from nose / body length.

Now, the only problem with this is that no one seems to be able to carry out this integration symbolically. This includes the powerful symbolic math aids. However, it is rather straightforward to perform the integration numerically and I have a numerics page that describes the mathematical and programming techniques. For those of who who just want the answer, get the factor from the following table for the fineness ratio (length/max diameter) of your body.

f | factor |
---|---|

2 | 0.7811 |

4 | 0.7358 |

6 | 0.7265 |

8 | 0.7232 |

10 | 0.7217 |

12 | 0.7208 |

14 | 0.7203 |

16 | 0.7200 |

18 | 0.7197 |

20 | 0.7196 |

22 | 0.7195 |

24 | 0.7194 |

26 | 0.7193 |

28 | 0.7192 |

30 | 0.7192 |

40 | 0.7191 |

50 | 0.7190 |

Then compute the surface area from

$$S=(2\pi {r}_{\mathrm{max}}L)\mathrm{factor}$$The quantity in parentheses is the surface area of a cylinder of the same length and diameter as the Sears-Haack body. So, if you just remember that the area is about 72 percent of that, that will be good enough for most purposes.