How do you "read" this formula?

[MyNameIsNicholas]

Homework Statement

No actual work, could just use some assistance in understanding formulas involving the centroid of an object, specifically with integrals. For example, how would you understand the following formula(s) (as seen in part 2)? I understand that the centroid is the sum of all the centroids of its smaller shapes (quadrilaterals, triangles, other simple geometrical shapes, etc.) divided by the total area (2-D).

This same formula (or a similar variant) can be used to also find centre of gravity, centroid of a line, 3-D object (volume), etc.

My primary confusion comes from the integral without an upper bound. How is that perceived / computed (in general given that I have not provided a specific problem).

Homework Equations

The Attempt at a Solution

N/A

Thank you, any help is much appreciated

 

[phyzguy]

When the integral is written like that, it means the integral over the entire area A. It is a shorthand which is meant to apply regardless of the shape of the area A. In order to actually do the integral, it is up to you to figure out the limits on x and y which cover the whole area, and to figure out the area element dA. In rectangular coordinates, dA = dx*dy, but in other coordinate systems it will be different.

 

[haruspex]

My primary confusion comes from the integral without an upper bound.

The subscript A indicates an integral over the area A, whatever size and shape that is. It is effectively a double integral, but you get to choose the coordinates and how to express the bounds. For a rectangular region, no doubt you would choose Cartesian coordinates, rotated as necessary to align with the boundaries. For complicated shapes you can break it into a sum of integrals, maybe using different coordinates in each piece.