Challenging volume

魏王將張遼

2009-05-18 11:42 pm

A circluar hole with radius r is drilled through a sphere of radius R with the separation betweeen the centres being d as shown in the diagram below:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/May09/CrazyQ3.jpg

By considering a double integral, or otherwise, find the volume of the drilled part.
Note: If it can be proved that there's no closed form, answer in approximation form will be acceptable.

回答 (1)

✔ 最佳答案

如圖:

圖片參考：http://s585.photobucket.com/albums/ss296/mathmanliu/DrilledSph.gif

或參考:http://www.wretch.cc/album/show.php?i=mathmanliu&b=1&f=1740510797&p=116
設OA= a, OE=R, AB=x=AC
By shell method, "shell" DEFG以BC切開上下兩個重新組合
得 new shell, 半徑AB=x, 高=CD+BE=√[R^2-(a+x)^2] + √[R^2-(a-x)^2]
總體積=∫[0~r] 2πx{R^2-(a+x)^2] + √[R^2-(a-x)^2]} dx

圖片參考：http://s585.photobucket.com/albums/ss296/mathmanliu/DrilledSph_sol.gif

或參考 http://www.wretch.cc/album/show.php?i=mathmanliu&b=1&f=1740510798&p=117