# Finding the centroid of the region

• thekey
In summary, to find the centroid of the region bounded by the curves y = 2x - 4 , y = 2 Sqr x, and x = 1, we need to find the area of the region and then calculate the x and y coordinates of the centroid using the integrals given above. This can be done by solving for the points of intersection between the curves, finding the area with a single or double integral, and then using the formulas for the x and y coordinates of the centroid.
thekey
Hi

VVVVVV

Find the centroid of the region bounded by the curves y = 2x - 4 , y = 2 Sqr x, and x = 1. Make a sketch of the region.

Make an attempt?

I did not try ..because I am not familiar with the way of solving

Did you at least manage to sketch the region? You have three lines. Find out where they meet.

yeah I did sketched the three curves then ?..

So where do they intersect? What formula do you know for finding a coordinate of a centroid?

they intersect between 1 and ( a unknown point intersection between 2 Sqrt x and x = 1 )

also, there is a triangle region under x-axis between 1 and 2 but I think it is not included in the intersection

how can I know the second point of the limit integration ?!

I got the answer :D >>

thank u : )

thekey said:
Hi

VVVVVV

Find the centroid of the region bounded by the curves y = 2x - 4 , y = 2 Sqr x, and x = 1. Make a sketch of the region.
I am puzzled by this. Why in the world would you be given a homework problem like this if you had never been given instruction in these and your textbook has nothing on it? You are taking Calculus are you not? And every text I have seen has the formulas for "centroid". You also seem to be saying that you cannot solve a simple quadratic equation.

In any case, The line x= 1 intersects $y= 2\sqrt{x}$ at (1, 2) and the line y= 2x- 4 at (1, -2) and forms the left boundary. The line y= 2x- 4 and $y= 2\sqrt{x}$ intersect when $y= 2x- 4= 2\sqrt{x}$. Divide by 2 to get $x- 2= \sqrt{x}$ and square both sides: $(x- 2)^2= x^2- 4x+ 4= x$ or $x^2- 5x+ 4= 0$. That factors easily: (x- 4)(x- 1)= 0. Either x= 1 or x= 4. The point (1, -4) is an "extraneous" root- it is not on $y= 2\sqrt{x}$. So the last vertex, the intersection between $y= 2\sqrt{x}$ and y= 2x- 4, is at (4, 4).

The area of that region is given by the single integral
$$\int_1^4 2\sqrt{x}- (2x- 4)dx$$
which could also be done by the double integral
$$\int_1^4\int_{2x- 4}^{2\sqrt{x}} dydx$$.

I give that double integral (which easily integrates with respect to y to give the first integral) because it is needed for the centroid.

The x coordinate of the centroid is given by the integral of x over that region
$$\int_1^4\int_{2x-4}^{2\sqrt{x}} x dydx= \int_1^4 x(2\sqrt{x}- (2x-4))dx$$
divided by the area and

the y coordinate of the centroid is given by the integral of y over that region
$$\int_1^4\int_{2x-4}^{2\sqrt{x}} y dy dx$$
divided by the area.

## 1. What is the centroid of a region?

The centroid of a region is the geometric center or average position of all the points in the region. It is often referred to as the center of mass or balance point of the region.

## 2. How is the centroid of a region calculated?

The centroid of a region can be calculated by finding the weighted average of all the points in the region, with each point's weight being its respective area or volume. This can be done using mathematical formulas or by dividing the region into simpler shapes and finding the centroids of each shape.

## 3. What is the significance of finding the centroid of a region?

The centroid of a region is significant because it can help in determining the stability, balance, and symmetry of the region. It is also useful in various engineering and scientific applications, such as calculating the center of pressure in fluid mechanics or determining the center of gravity in mechanics.

## 4. Can the centroid of a region be outside of the region?

Yes, the centroid of a region can be outside of the region. This can happen when the region is irregularly shaped or has varying densities. In such cases, the centroid may lie outside the region, but it still represents the average position of all the points in the region.

## 5. How does the centroid of a 2D region differ from that of a 3D region?

In a 2D region, the centroid is a single point with x and y coordinates, while in a 3D region, the centroid is a point with x, y, and z coordinates. The calculation methods for both are also different, as a 3D region requires finding the volume of the region and using that as a weight in the calculation.

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