To find the centroid of the region bounded by the curves y = x³, y = 10, and y = 0, we first need to identify the area of the region and then calculate the coordinates of the centroid.
Step 1: Identify the Curves
- y = x³ is a cubic function that increases from the origin.
- y = 10 is a horizontal line.
- y = 0 is the x-axis.
Step 2: Find Intersection Points
To determine the region of interest, we need to find the points where the curves intersect. Set x³ = 10 to find where the cubic intersects the line:
x = 10^{1/3} > ext{approximately } 2.154So, the region we’re interested in is bounded between x = 0 and x = 10^{1/3}.
Step 3: Calculate the Area of the Region
The area A of the region can be calculated by integrating the difference between the upper function (y = 10) and the lower function (y = x³):
A = ∫_{0}^{10^{1/3}} (10 - x^{3}) \, dxEvaluating this integral gives:
A = [10x - (1/4)x^{4}]_{0}^{10^{1/3}}Calculating this from 0 to 10^{1/3} yields:
A = 10(10^{1/3}) - (1/4)(10^{4/3}) = 10^{4/3} - (1/4)(10^{4/3}) = (3/4)(10^{4/3})Step 4: Calculate Centroid Coordinates
The x-coordinate of the centroid (ar{x}) can be found using the formula:
ar{x} = (1/A) ∫_{a}^{b} x(10 - x^{3}) \, dxwhere a = 0 and b = 10^{1/3}. After integrating and simplifying, you can find ar{x}. Similarly, to find the y-coordinate of the centroid (ar{y}):
ar{y} = (1/A) ∫_{a}^{b} (1/2)(10 - x^{3})^{2} \, dxAfter performing both integrations, you will arrive at numeric values for both ar{x} and ar{y}.
Conclusion
The centroid coordinates (ar{x}, ar{y}) provide the center of mass for the bounded region, which is essential for many applications in physics and engineering. By following the above steps meticulously, you can accurately determine the centroid for the given curves.