球的体积公式推导

球的体积公式推导

作者：追风剑情发布于：2022-8-10 11:30 分类：Algorithms

设：球的体积V，半径R，将球划分成n个圆柱体，每个圆柱体的半径为r，高为h(微元d_h)，底面积为s，体积为v，有

\begin{aligned} h = R \cdot s i n θ \\ r = R \cdot c o s θ \\ s = π r^{2} \\ v = s \times h = π r^{2} \times R \cdot s i n θ = π (R \cdot c o s θ)^{2} \times R \cdot s i n θ \end{aligned}

\begin{aligned} V & = 2 \int_{0}^{R} s d_{h} = 2 \int_{0}^{R} π r^{2} d_{h} = 2 \int_{0}^{π / 2} π (R \cdot c o s θ)^{2} d_{(R \cdot s i n θ)} \\ = 2 π R^{3} \int_{0}^{π / 2} c o s^{2} θ d_{s i n θ} \\ = 2 π R^{3} \int_{0}^{π / 2} (1 - s i n^{2} θ) d_{s i n θ} \\ = 2 π R^{3} (\int_{0}^{π / 2} d_{s i n θ} - \int_{0}^{π / 2} s i n^{2} θ d_{s i n θ}) \\ = 2 π R^{3} {[s i n θ - \frac{1}{3} s i n^{3} θ]}_{0}^{π / 2} \\ = 2 π R^{3} (1 - \frac{1}{3}) \\ = \frac{4}{3} π R^{3} \end{aligned}

标签: Algorithms

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