翻訳と辞書 Words near each other ・ Gabriela Radice ・ Gabriela Rejala ・ Gabriela Reyes ・ Gabriela Rivadeneira ・ Gabriela Rivero ・ Gabriel Álvarez ・ Gabriel Álvarez de Toledo ・ Gabriel Álvez ・ Gabriel Ávalos ・ Gabriel Özkan ・ Gabriel Øidne ・ Gabriel's Dream ・ Gabriel's Fire ・ Gabriel's Gift ・ Gabriel's Gully ・ Gabriel's Horn ・ Gabriel's Inferno ・ Gabriel's Lament ・ Gabriel's Landing ・ Gabriel's Message ・ Gabriel's Mother's Highway ・ Gabriel's Oboe ・ Gabriel's Revelation ・ Gabriel's Story ・ Gabriel's theorem ・ Gabriel's Wharf ・ Gabriel's Wing ・ Gabriel, comte de Montgomery ・ Gabriel, Lord of Monaco ・ Gabriel-Auguste Ancelet Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Gabriel's Horn ： ウィキペディア英語版 Gabriel's Horn Gabriel's Horn (also called Torricelli's trumpet) is a geometric figure which has infinite surface area but finite volume. The name refers to the tradition identifying the Archangel Gabriel as the angel who blows the horn to announce Judgment Day, associating the divine, or infinite, with the finite. The properties of this figure were first studied by Italian physicist and mathematician Evangelista Torricelli in the 17th century. ==Mathematical definition== Gabriel's horn is formed by taking the graph of $x\; \backslash mapsto\; \backslash frac$, with the domain $x\; \backslash ge\; 1$ (thus avoiding the asymptote at ''x'' = 0) and rotating it in three dimensions about the x-axis. The discovery was made using Cavalieri's principle before the invention of calculus, but today calculus can be used to calculate the volume and surface area of the horn between ''x'' = 1 and ''x'' = ''a'', where ''a'' > 1. Using integration (see Solid of revolution and Surface of revolution for details), it is possible to find the volume $V$ and the surface area $A$: :$V\; =\; \backslash pi\; \backslash int\_^\; \backslash left(\; \backslash right)^2\backslash ,\; \backslash mathrmx\; =\; \backslash pi\; \backslash left(\; 1\; -\; \backslash right)$ :$A\; =\; 2\backslash pi\; \backslash int\_^\; \backslash sqrt\backslash ,\backslash mathrmx\; >\; 2\backslash pi\; \backslash int\_^\; \backslash ,\backslash mathrmx\; =\; 2\backslash pi\; \backslash ln\; a.$ $a$ can be as large as required, but it can be seen from the equation that the volume of the part of the horn between $x\; =\; 1$ and $x\; =\; a$ will never exceed $\backslash pi$; however, it ''will'' get closer and closer to ''$\backslash pi$'' as $a$ becomes larger. Mathematically, the volume ''approaches ''$\backslash pi$'' as $a$ approaches infinity''. Using the limit notation of calculus: :$\backslash lim\_V\; =\; \backslash lim\_\backslash pi\; \backslash left(\; 1\; -\; \backslash right)\; =\; \backslash pi.$ The surface area formula above gives a lower bound for the area as $2\backslash pi$ times the natural logarithm of $a$. There is no upper bound for the natural logarithm of $a$ as $a$ approaches infinity. That means, in this case, that the horn has an infinite surface area. That is to say; :$\backslash lim\_A\; \backslash ge\; \backslash lim\_2\; \backslash pi\; \backslash ln\; a\; =\; \backslash infty.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Gabriel's Horn」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.