How do you find the centroid of the region bounded by the curves y = x^3, y = x, y = 2, and y = 0?

Finding the Centroid of Bounded Curves

To find the centroid (center of mass) of the region bounded by the curves:

• y = x³
• y = x
• y = 2
• y = 0

We can follow these steps:

Step 1: Determine the Points of Intersection

First, we need to find the intersection points of the curves:

• Set y = x and y = x³:
  
  x = x³
  
  x³ – x = 0
  
  x(x² – 1) = 0
  
  x = 0, x = 1, and x = -1 (but we are interested only in the region of interest where y is positive).
• Set y = x and y = 2:
  
  x = 2

So, the points of intersection that define the region are at:

• (0, 0)
• (1, 1)
• (2, 2)

Step 2: Set Up the Integrals

To find the area of the region and the coordinates of the centroid (x_c, y_c), we use:

Area (A)

The area A can be found using:

		A = ∫[f(x) - g(x)] dx
		

where f(x) is the upper curve (y = 2) and g(x) is the lower curve (y = x³) in the interval from 0 to 1.

Centroid Coordinates

The coordinates of the centroid can be calculated using the formulas:

• x_c = (1/A) ∫ x(f(x) – g(x)) dx
• y_c = (1/A) ∫ (1/2)(f(x)² – g(x)²) dx

Where the limits of integration will be defined based on the region from 0 to 2.

Step 3: Perform the Integration

 

1. First, calculate the area:

		A = ∫02(2 - 0) dx = 2 
	

2. Now calculate x_c:

		xc = (1/A) ∫02x(2 - 0) dx 
		 = (1/2) ∫022x dx 
		 = (1/2)[x2]|02 
		 = 2/3 
	

3. Finally, calculate y_c:

		yc = (1/A) ∫02(1/2)(4 - 0) dx = (1/2)(4) = 2 
	

Final Result

The coordinates of the centroid of the bounded region are:

• x_c = 2/3
• y_c = 2

Therefore, the centroid of the region bounded by the given curves is at the point (2/3, 2).