# Step 1: **Find the Coordinates \((x_m, y_m)\) of the Shaded Area** ## **Explanation:** The shaded region is inside a square of side \(2r\), bounded by: - The square (from \(x=\) to \(x=2r\), \(y=\) to \(y=2r\)) - The quarter circle (center at \((r, r)\), radius \(r\)) - The triangle below the quarter circle The centroid \((x_m, y_m)\) of the shaded area is given: \[ x_m = y_m = \frac{4r}{3\pi} \] (As given in the problem.) ### **Why?** For a quarter circle of radius \(r\), the centroid from the straight edges is \(\frac{4r}{3\pi}\). Since the shaded area is the quarter circle minus the triangle, and the centroid formula provided matches the quarter circle, we use this value. --- # Step 2: **Find the Volume Formed by Rotating the Shaded Area Around the \(x\)-axis** ## **Explanation:** The shaded area is a **quarter circle** (area \(A_1\)) **minus a triangle** (area \(A_2\)). ### **Area Calculations:** - **Quarter circle:** \[ A_1 = \frac{1}{4} \pi r^2 \] - **Triangle (right triangle with legs \(r\)):** \[ A_2 = \frac{1}{2} r \cdot r = \frac{r^2}{2} \] - **Shaded area:** \[ A = A_1 - A_2 = \frac{1}{4} \pi r^2 - \frac{r^2}{2} \] ### **Volume of Revolution (about \(x\)-axis):** When the shaded area is revolved about the \(x\)-axis, the volume is: \[ V = \text{Shaded Area} \times (\text{Circumference traced by centroid}) \] The centroid's distance from the x-axis is \(y_m\), so the path traced is a circle of radius \(y_m\): \[ V = (\text{Shaded Area}) \times (2\pi y_m) \] Plugging in the values: - \(y_m = \frac{4r}{3\pi}\) - \(\pi = 3\) (as given) Calculate the shaded area: \[ A = \frac{1}{4} \cdot 3 \cdot r^2 - \frac{r^2}{2} = \frac{3}{4} r^2 - \frac{1}{2} r^2 = \frac{3-2}{4} r^2 = \frac{1}{4} r^2 \] Now plug into the volume formula: \[ V = \frac{1}{4} r^2 \cdot 2\pi \cdot \frac{4r}{3\pi} \] \[ V = \frac{1}{4} r^2 \cdot \frac{8r}{3} \] \[ V = \frac{8}{12} r^3 = \frac{2}{3} r^3 \] --- # **Final Answer** - **Centroid coordinates:** \(x_m = y_m = \frac{4r}{3\pi}\) - **Volume of solid formed:** \[ \boxed{V = \frac{2}{3} r^3} \]