Given the region bounded above by $y=x^2$ and below by $y=x$ on the interval $0\le x\le 1$, determine the $x$- and $y$-coordinates of the centroid of the shaded area.

• A. $x=0.7$, $y=0.25$
• B. $x=0.6$, $y=0.35$
• C. $x=0.583$, $y=0.283$
• D. $x=0.5$, $y=0.5$

Answer: (C)

Verified step by step guidance

1. Step 1: Understand the problem. The centroid of a region is the 'center of mass' of the area. To find the centroid, we need to calculate the x-coordinate (denoted as \( \bar{x} \)) and y-coordinate (denoted as \( \bar{y} \)) using formulas for the centroid of a region bounded by curves.
2. Step 2: Recall the formulas for the centroid. The x-coordinate of the centroid is given by \( \bar{x} = \frac{1}{A} \int_{a}^{b} x \cdot [f(x) - g(x)] \, dx \), and the y-coordinate is given by \( \bar{y} = \frac{1}{A} \int_{a}^{b} \frac{[f(x) + g(x)]}{2} \cdot [f(x) - g(x)] \, dx \), where \( A \) is the area of the region, \( f(x) \) is the upper curve, and \( g(x) \) is the lower curve.
3. Step 3: Calculate the area \( A \) of the region. The area is given by \( A = \int_{a}^{b} [f(x) - g(x)] \, dx \). Here, \( f(x) = x^2 \) and \( g(x) = x \), and the interval is \( [0, 1] \). Set up the integral \( A = \int_{0}^{1} (x^2 - x) \, dx \). Evaluate this integral to find \( A \).
4. Step 4: Set up the integral for \( \bar{x} \). Substitute \( f(x) = x^2 \) and \( g(x) = x \) into the formula for \( \bar{x} \): \( \bar{x} = \frac{1}{A} \int_{0}^{1} x \cdot (x^2 - x) \, dx \). Simplify the integrand and evaluate the integral.
5. Step 5: Set up the integral for \( \bar{y} \). Substitute \( f(x) = x^2 \) and \( g(x) = x \) into the formula for \( \bar{y} \): \( \bar{y} = \frac{1}{A} \int_{0}^{1} \frac{(x^2 + x)}{2} \cdot (x^2 - x) \, dx \). Simplify the integrand and evaluate the integral. Combine the results to find the coordinates of the centroid.