A First Course in String Theory

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Show that

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**Geometric proof**

The relations and and thus the volumes of -balls and areas of -spheres can also be derived geometrically. As noted above, because a ball of radius is obtained from a unit ball by rescaling all directions in times, is proportional to , which implies .

Also, because a ball is a union of concentric spheres and increasing radius by corresponds to a shell of thickness . Thus, ; equivalently, .

— Wikipedia on *Volume of an -ball*

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— Me@2022-05-18 09:08:11 AM

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2022.05.19 Thursday (c) All rights reserved by ACHK