# Second moment of surface area of cone

By dividing the surface into ring elements, find the second moment, about the axis of symmetry, of a conical surface of slant length $l$ and semi-vertical angle $30^o$
If the cone is represented by the figure generated when the line $y=\frac{\sqrt{3}}{3} x$ is rotated completely around the x-axis.
$\displaystyle M=2\pi\int^a_b y^3 dx$
$\displaystyle =2\pi\int^{\frac{\sqrt{3}}{2}l}_0 \frac{\sqrt{3}}{9}x^3$
$\displaystyle = \frac{\sqrt{3}}{32}\pi l^4$
Answer is $\frac{1}{16}\pi l^4$