I essentially have the same question, however only concerning the First moment, as this user ,however none of the answers satisfy my questions.
I understand that we are trying to get a sense of the area distribution of an object. I do not understand how that is achieved using this equation $$\int x dA $$
multiplying the area by the distance from your respective origin, how does that give the distribution? We get units of volume so that is even more confusing.
For example suppose we are trying to find the centroid of a triangle of b = 7 and h =11.
when finding the y coordinate of the centroid $$(\int y/2 dA)/\int dA $$
where we do y/2 as the centroid for each rectangle of the differential is at y/2. However i do not understand the intuition behind that, why do we want to multiply the center for each differential by its area? and why does dividing by the area give us the y coordinate of the centroid of the triangle.
All the answers so far have been the same where they do not really explain why it is fundamentally, I am hoping someone can help me out.