【Math】高數-一個有趣的旋轉體體積與面積

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題目
設曲線 \(y = \tfrac{1}{x}\) (1\(\leq\)x\(\leq\)+\(\infty\)) 與X 軸之間的區域為D.
求D繞X軸旋轉一周的旋轉體的體積V和面積A。
圖形如下

畫出圖形容易計算體積

\[ V = \begin{aligned} \int_1 ^{+\infty} \pi ( \tfrac{1}{x} )^2 \mathrm{d} x \end{aligned} = \pi \]

體積公式

\[ A = \int_1^{+\infty} 2 \pi y \sqrt{1+(y^{'})^2} \mathrm{d} x = 2 \pi \int_1^{+\infty} \tfrac{1}{x} \sqrt{1+\tfrac{1}{x^4}}\mathrm{d} x \geq 2 \pi \int_1^{+\infty} \tfrac{1}{x} \mathrm{d} x = {+\infty} \]

即體積V = \(\pi\) ,表面積A = \(+\infty\)
容積有限，表面積卻無限，
Gabriel's horn (also called Torricelli's trumpet) is a geometric figure which has infinite surface area but finite volume.
也就是說往這個喇叭中裝滿油漆，但是里面的油漆卻無法將喇叭的內側塗一遍。
Since the horn has finite volume but infinite surface area, there is an apparent paradox that the horn could be filled with a finite quantity of paint and yet that paint would not be sufficient to coat its inner surface.
這悖論挺有意思的，
The paradox is resolved by realizing that a finite amount of paint can in fact coat an infinite surface area — it simply needs to get thinner at a fast enough rate.
這是Gabriel's Horn，聖經中天使之一，上帝傳送好消息給人類的使者。