Centroid of homogeneous lamina region R and the factor of "1/2"

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Summary:

I found the formulas in two different sources for y-coordinate of center of gravity for a homogeneous lamina differ from each other by a factor "1/2".
Hi,

In one of the standard calculus textbooks, source #1, the formula for y-coordinate of center of gravity for a homogeneous lamina is given as follows.

?hash=f2d1f9b2061ada94c12949155edcf0e6.jpg


In another book of formulas, source #2, the formula is given without the factor "1/2" as is shown below. Personally, I believe that source #1 is correct. I just wanted to confirm so that I could notify the publisher of source #2 of erratum. Could you please help me with it? Thank you!

?hash=f2d1f9b2061ada94c12949155edcf0e6.jpg
 

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  • #2
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Hi,

Could someone please comment on it? Thank you!
 
  • #3
pasmith
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If you start with the area integral for the centroid of the region bounded by [itex]x = a[/itex], [itex]x = b[/itex], [itex]y = 0[/itex] and [itex]y = f(x)[/itex] you get [tex]
\bar y = \frac{\int_A y\,dA}{\int_A\,dA} = \frac{\int_a^b \int_0^{f(x)} y\,dy\,dx}{\int_a^b \int_0^{f(x)}\,dy\,dx} = \frac{\int_a^b \left[ \frac12 y^2 \right]_0^{f(x)}\,dx}{\int_a^b \left[y\right]_0^{f(x)}\,dx}
= \frac{\int_a^b \frac12 (f(x))^2\,dx}{\int_a^b f(x)\,dx}[/tex] as stated.
 
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  • #4
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Thank you!

So, the factor of "1/2" should be there. I will let the publisher of source #2 that there is an erratum. Thanks.
 
  • #5
pasmith
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Thank you!

So, the factor of "1/2" should be there. I will let the publisher of source #2 that there is an erratum. Thanks.
The formula given in source #2 is the correct formula: by definition [itex]\bar y = \frac1A \int_A y\,dA[/itex].

The error in source #2 is not adequately explaining how to calculate it. It's only a coincidence that setting [itex]dA = f(x)\,dx[/itex] gives the correct answer for [itex]\int x\,dA[/itex]; doing the same for [itex]\int y\,dA[/itex] does indeed result in the answer being out by a factor of two. I don't think setting [itex]dA = g(y)\,dy[/itex] assists; after all, what is [itex]g[/itex] if [itex]f[/itex] is not strictly monotonic?
 
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