By dividing the surface into ring elements, find the second moment, about the axis of symmetry, of a conical surface of slant length $\displaystyle l$ and semi-vertical angle $\displaystyle 30^o$

If the cone is represented by the figure generated when the line $\displaystyle y=\frac{\sqrt{3}}{3} x$ is rotated completely around the x-axis.

The formula for the second moment

$\displaystyle \displaystyle M=2\pi\int^a_b y^3 dx$

$\displaystyle \displaystyle =2\pi\int^{\frac{\sqrt{3}}{2}l}_0 \frac{\sqrt{3}}{9}x^3$

$\displaystyle \displaystyle = \frac{\sqrt{3}}{32}\pi l^4$

Answer is $\displaystyle \frac{1}{16}\pi l^4$

Thanks!