Find the second moment, about the specified line, of each of the following areas:

A triangle ABC in which $\displaystyle \angle ABC = 90^{o}$ and $\displaystyle AB=BC=l$, about AB.

I used $\displaystyle M=x^2\delta A$ for this.

The triangle is represented by the axes and the line $\displaystyle y=l-x$.

With AB vertical.

$\displaystyle \delta A=x\delta y$

$\displaystyle M=x^3\delta y$

So the second moment of area about the line AB

$\displaystyle \displaystyle =\int^l_0 (l-y)^3dy$

$\displaystyle \displaystyle =\int^l_0 (l^3-3l^2y+3ly^2-y^3)dy$

$\displaystyle \displaystyle = \left[l^3y-\frac{3}{2}l^2y^2+ly^3-\frac{y^4}{4}\right]^l_0$

$\displaystyle =\frac{1}{4}l^4$

Answer is $\displaystyle \frac{1}{12}l^4$

Thanks!