Please explain why "1/2" appears in the following summation.
My understanding is that Mx = y*(dA), so I'm looking for an interpretation of the 1/2 factor there especially, as for y=f(x) and A=f(x)*dx it seems logical to have f(x) squared, but what is the geometrical reason to bring the 1/2 factor in?
The final goal is to come to the integral formula that calculates Mx:
The centre of mass of a uniform rectangle is in the centre of the rectangle. If the bottom edge of the rectangle is the $x$ axis, the centre is at $\frac12 y$. The distance to the centre of mass is multiplied by the "mass" of the rectangle, its area $y \times \Delta x$, to give $\frac12 y\times y\times \Delta x$.
The whole calculation is the centroids of the rectangles defined by $f$ about the $x$ axis minus the centroids of the rectangles defined by $g$ about the $x$ axis.