Question

1. In this problem, we will investigate the Paradox of Gabriel's Horn: There exists a hollow solid that can be completely filled with a finite amount of paint, and yet no amount of paint will be sufficient to cover its entire surface. Let S be the surface (Gabriel's Horn) obtained by rotating the curve yaround the r-axis for 2 1, and let Sb be the surface obtained by rotating the curve y- around the r-axis for 1 3Sb. IfV and V, are the volumes enclosed by S and S, and A and Ab are the surface areas of S and Sb, then V=·lim Vs A = 'lim Ab (a) Use cylindrical coordinates centered around the r-axis to compute the volume V% of S (b) Find a parametric representation r(u, v) for S, and use it to compute the surface area Ab of S c) Compute Vlim V and A-lim Ab. Explain why the values of V and A seem to be a paradox

Answer 1

Solution:

a)

Gabriel's horn is given by the equation

2 2

The above equation is true when y =-is rotated around x axis

So, in cylindrical coordinates y r cos θ, r sin θ.r--

$\text{V}=\int_{0}^{2\pi}\int_{\frac{1}{b}}^{1}\int_1^{\frac{1}{r}}\mathrm{d}x r\mathrm{d}r\mathrm{d}\theta$

$\text{V}=\int_{0}^{2\pi}\int_{\frac{1}{b}}^{1}\left [ x \right ]_1^{\frac{1}{r}}r\mathrm{d}r\mathrm{d}\theta$

$\text{V}=\int_{0}^{2\pi}\int_{\frac{1}{b}}^{1}\left ( \frac{1}{r}-1 \right )r\mathrm{d}r\mathrm{d}\theta$

(1) drde 0

2 0

(b - 1)2 V=2n 262

$\text{V}= \dfrac{\pi\left(b-1\right)^2}{b^2}$