欢迎光临散文网会员登陆 & 注册

[Calculus] Volume of the Tricylinder

2021-07-19 08:27 作者:AoiSTZ23 0人读过 | 我要投稿

By: Tao Steven Zheng (郑涛)

【Problem】

The tricylinder is a solid formed by intersecting three perpendicular cylinders:

$x%5E2%20%2B%20y%5E2%20%3D%20r%5E2%20$

$x%5E2%20%2B%20z%5E2%20%3D%20r%5E2%20$

$y%5E2%20%2B%20z%5E2%20%3D%20r%5E2%20$

Determine the volume of the tricylinder.

【Solution】

Dissect the tricylinder into seven pieces: 1 cube of side length $%5Csqrt%7B2%7Dr$ , and 6 congruent “caps“. The integral for calculating the volume of the cap involves integrating the square cross-sections $4(r%5E2%20-%20x%5E2)%20$ on the interval $r%2F%5Csqrt%7B2%7D%20%3C%20x%20%3C%20r$ .

The volume of the cube is

$%7BV%7D_%7Bcube%7D%20%3D%20%7B%5Cleft(%5Csqrt%7B2%7D%20r%20%5Cright)%7D%5E%7B3%7D$

The volume of each cap is

$%7BV%7D_%7Bcap%7D%20%3D%20%5Cint_%7Br%2F%5Csqrt%7B2%7D%7D%5E%7Br%7D%204(r%5E2%20-%20x%5E2)%20dx$

$%7BV%7D_%7Bcap%7D%20%3D%204%5Cleft%5Br%5E2x-%5Cfrac%7Bx%5E3%7D%7B3%7D%5CBiggr%7C_%7Br%2F%5Csqrt%7B2%7D%7D%5E%7Br%7D%5Cright%5D$

$%7BV%7D_%7Bcap%7D%20%3D%20%5Cfrac%7B1%7D%7B3%7D%5Cleft(8-5%5Csqrt%7B2%7D%5Cright)%7Br%7D%5E%7B3%7D$

Thus, the volume of the tricylinder is

$V%20%3D%20%7BV%7D_%7Bcube%7D%20%2B%206%7BV%7D_%7Bcap%7D$

$V%20%3D%202%5Csqrt%7B2%7D%20%7Br%7D%5E%7B3%7D%20%2B%206%5Ctimes%5Cfrac%7B1%7D%7B3%7D%5Cleft(8-5%5Csqrt%7B2%7D%5Cright)%7Br%7D%5E%7B3%7D%20$

$V%20%3D%20%5Cleft(16-8%5Csqrt%7B2%7D%5Cright)%7Br%7D%5E%7B3%7D$

标签：

[Calculus] Volume of the Tricylinder的评论 (共条)