# Find the centroid of the region

• dohsan
In summary, the centroid of a region is the point where the mass of the region is evenly distributed and is often referred to as the geometric center or balance point. The formula for finding the centroid is (x̄, ȳ) = (∫xρdA/∫ρdA, ∫yρdA/∫ρdA), and it can lie outside of the region in irregularly shaped or non-uniform regions. The centroid is used in various real-world applications such as engineering, physics, and geography to find the center of mass, determine stability, and locate the center of population. It can change if the distribution of mass or shape of the region changes, but remains fixed for regular shapes
dohsan

## Homework Statement

Find the centroid of the region bounded by ...

y=x^3, x+y=2, y=0

## The Attempt at a Solution

I think you find the area of the region which is 5/4

Sure, that's the area alright. What else do you have to compute to find the centroid?

That is NOT what I get for the area! dohsan, could you show your work?

## 1. How is the centroid of a region defined?

The centroid of a region is defined as the point where all the mass of the region is evenly distributed. It is the geometric center or the balance point of the region.

## 2. What is the formula for finding the centroid of a region?

The formula for finding the centroid of a region is (x̄, ȳ) = (∫xρdA/∫ρdA, ∫yρdA/∫ρdA), where x and y are the coordinates of the centroid, ρ is the density of the region, and A is the area of the region.

## 3. Can the centroid of a region lie outside of the region?

Yes, the centroid of a region can lie outside of the region. This is possible in cases where the region is irregularly shaped or has a non-uniform distribution of mass.

## 4. How is the centroid of a region used in real-world applications?

The centroid of a region is used in various real-world applications such as engineering, physics, and geography. It is used to find the center of mass of an object, determine the stability of structures, and locate the center of population in a geographical region.

## 5. Can the centroid of a region change?

Yes, the centroid of a region can change if the distribution of mass within the region changes. A shift in the position of mass or a change in the shape of the region can cause the centroid to move. However, for a regular shape with a uniform distribution of mass, the centroid will remain fixed.

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