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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?

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The final goal is to come to the integral formula that calculates Mx: enter image description here

user551034
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  • I think this needs more context to be able to answer it. – Suzu Hirose Jul 28 '22 at 01:01
  • @Suzu Hirose, the title is "Centroid of a plane area", a definition, so there is no content actually. Say, if to calculate dMy, we have My = xdA, i.e. integrating x[f(x)-g(x)]dx from a to b, obviously. It is difficult to interpret Mx though. – user551034 Jul 28 '22 at 01:23
  • Maybe since it is showing f(xr*)^2/2 it has something to do with the integral of x being x^2/2? – Andrew Carratu Jul 28 '22 at 01:29
  • I agree @AndrewCarratu, but it is prior to taking into integral, it is still the summation, so I'm trying to understand how it reasons to form the summation. There must be some geometrical reasoning similar to when calculating My. – user551034 Jul 28 '22 at 01:31
  • It doesn't need reasoning, if you talk about moment about an axis it sounds like you have rotated the thing around the axis. If you had said you wanted the centroid of a plane area from the beginning it would have been easy to answer this question. – Suzu Hirose Jul 28 '22 at 01:46

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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.

Suzu Hirose
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