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While finding center of mass of extended bodies, we generally write the coordinate of the starting point of the infinitesimal mass element. Why not the ending or starting point? How does the error while writing the coordinates gets resolved in integration?

Reference: Concepts of Physics by H.C. Verma, page 141 (at the end of the first paragraph under center of mass of a uniform straight rod)

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    $\begingroup$ @Mechanic I think OP means whether there is a difference in taking the left, right or center of an infinitesimal element $\text{d}x$ to be its position. $\endgroup$
    – Vincent Thacker
    Commented Sep 28, 2021 at 13:53

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The process of integration will take the limit as the size of the infinitesimal element goes to zero. In this limit, the starting and ending points of any element become the same. There is no error introduced by using the starting, ending, center, or any other point in a mass element as its coordinate.

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  • $\begingroup$ If doing a numeric integration (for a more complex shape) this becomes a valid concern. You would want the distance to be measure to the center of your (finite sized) mass element. $\endgroup$
    – R.W. Bird
    Commented Sep 28, 2021 at 18:02

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