# Finding the y Component of the Centroid: Is My Reasoning Valid?

• Miike012
In summary, the test question asked for the y component of the centroid for a region bounded by the functions √x and x, given the density and mass of the region. The solution involved finding y bar by integrating the moment about x over total mass, as the y component of the centroid does not depend on the values of density and mass. The y coordinate of the centroid is independent of the density, and only changes if the density or mass is shifted.
Miike012
The test question was...
We are given a region bounded by the functions √x and x. We were also given the density and mass of the region and asked to only the component y of the centroid.

I did not know how to implement these given variables, density and mass into my equation for y bar so I found y bar by integrating the moment about x over total mass.

Is my reasoning valid... the y component of the centroid of the bounded region does not depend on the value of rho or mass given because if density is dispersed evenly through out the region (which it is or atleast it should be from what the book says) then y bar will be at the same location always. The only time it would move is if the density or mass was shifted.

Are you sure you were asked for the centroid?
The centroid is the 'geometric center' it doesn't depend on mass density at all.

If it's the center of mass then, you'll have to refresh my memory here, moment is used to describe pretty much everything so I'm not quite sure what you mean by moment, nor do I know what you mean by y bar.

Imagine a group of discrete objects of equal mass, how would their density and total mass change the value of the center of mass?

If it was the centroid then the density doesn't play a role at all and indeed the y coordinate of the centroid would be independant of the density.

## 1. How do I find the y-component of the centroid?

To find the y-component of the centroid, you can use the formula: y̅ = ∑(yiAi) / ∑Ai, where yi represents the y-coordinate of each point and Ai represents the area associated with that point. Alternatively, you can divide the sum of the products of each point's y-coordinate and its corresponding area by the total area of the shape.

## 2. Is my reasoning valid for finding the y-component of the centroid?

Yes, as long as you use the correct formula and follow the proper steps, your reasoning for finding the y-component of the centroid will be valid.

## 3. What if my shape has holes or cut-outs?

If your shape has holes or cut-outs, you will need to calculate the centroid of each individual section and then use the weighted average method to find the overall centroid for the entire shape.

## 4. Can I use the same method to find the centroid of any shape?

Yes, the formula for finding the centroid applies to all shapes, as long as you use the correct values for each point's coordinates and associated area.

## 5. How accurate is the centroid calculation?

The centroid calculation is considered to be highly accurate, as it takes into account the distribution of weight within a shape. However, the accuracy may be affected if the shape is irregular or has complex geometry.

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