This paper presents a systematic design approach for low-profile UWB body-of-revolution (BoR) monopole antennas with specified radiation objectives and size constraints. The proposed method combines a random walk scheme, the genetic algorithm, and a BoR moment method analysis for antenna shape optimization. A weighted global cost function, which minimizes the difference between potential optimal points and a utopia point (optimal design combining 3 different objectives) within the criterion space, is adapted. A 24'' wide and 6'' tall aperture was designed operating from low VHF frequencies up to 2 GHz. This optimized antenna shape reaches -15 dBi gain at 41 MHz on a ground plane and is only λ/12 in aperture width and λ/50 in height at this frequency. The same antenna achieves VSWR <; 3 from 210 MHz up to at least 2 GHz. Concurrently, it maintains a realized gain of ~5 dBi with moderate oscillations across the band of interest. A resistive treatment was further applied at the top antenna rim to improve matching and pattern stability. Measurements are provided for validation of the design. Of importance is that the optimized aperture delivers a larger impedance bandwidth as well as more uniform gain and pattern when compared to a previously published inverted-hat antenna of the same size.