Given the function $f\left(x\right)=x^2$ defined on the interval $0\le x\le 2$, determine the y-coordinate of the centroid of the region bounded by the graph of $f\left(x\right)$, the x-axis, and the lines $x=0$ and $x=2$.

• A. $1$
• B. $\frac{2}{3}$
• C. $\frac{8}{5}$
• D. $\frac{4}{3}$

Answer: (C)

Verified step by step guidance

1. Step 1: Recall the formula for the y-coordinate of the centroid of a region bounded by a curve, the x-axis, and vertical lines. The formula is: ȳ = (1/A) * ∫[a to b] (1/2) * [f(x)]^2 dx, where A is the area of the region, and f(x) is the function defining the curve.
2. Step 2: Compute the area (A) of the region. The area is given by A = ∫[a to b] f(x) dx. For this problem, f(x) = x^2, and the interval is [0, 2]. Set up the integral: A = ∫[0 to 2] x^2 dx.
3. Step 3: Solve the integral for the area. Use the power rule for integration: ∫x^n dx = (x^(n+1))/(n+1) + C. Apply this rule to ∫x^2 dx over the interval [0, 2].
4. Step 4: Substitute the computed area (A) into the formula for ȳ. Set up the integral for the numerator of ȳ: ∫[0 to 2] (1/2) * [f(x)]^2 dx = ∫[0 to 2] (1/2) * (x^4) dx. Use the power rule for integration to solve this integral.
5. Step 5: Divide the result of the numerator integral by the area (A) to find the y-coordinate of the centroid, ȳ. Simplify the expression to obtain the final value.