使用极限法推导球缺面积和体积公式
首发时间:2022-08-02
摘要:对球缺面积及体积公式的推导在之前的文献中往往采用微积分法或者其它方法间接得到,而本文提出一种运用极限思想仅使用初等数学知识,分别直接推导出球缺面积和体积公式的方法。推导过程中三角函数及图表消项的化简方法具有一定的通用参考性,且中间过程推导出的球内接多层堆叠圆台侧表面积及体积最简公式在需要离散求和时也可以直接应用,此外还讨论了使用圆柱近似的情形。
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Using the Limit Method to Derive the Formulas for the Area and Volume of a Spherical Cap
Abstract:The derivation of the spherical cap area and volume formulas is often obtained by calculus or other indirect methods in the previous literature, but this paper proposes a method to directly derive the area and volume formulas of the spherical cap by applying the limit idea only using elementary mathematics knowledge. The simplification methods of trigonometric function and chart elimination in the derivation process have certain general reference, and the simplest formulas of side surface area and volume of spherical inscribed multi-layer stacked conical frustum derived in the intermediate process can also be directly applied when discrete summation is required. In addition, the case of using cylindrical approximation is also discussed.
Keywords: spherical cap area spherical cap volume limit method conical frustum approximation cylindrical approximation
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