1. **Problem:** Find the centroid of the area bounded by the curves $y = 2x + 1$, $xy = 7$, and the vertical line $x = 8$. 2. **Step 1: Understand the boundaries** - The line is $y = 2x + 1$. - The hyperbola is $xy = 7$, or $y = \frac{7}{x}$. - The vertical boundary is $x = 8$. 3. **Step 2: Find points of intersection** - Find intersection of $y = 2x + 1$ and $y = \frac{7}{x}$: $$2x + 1 = \frac{7}{x} \implies 2x^2 + x - 7 = 0$$ - Solve quadratic: $$x = \frac{-1 \pm \sqrt{1 + 56}}{4} = \frac{-1 \pm \sqrt{57}}{4}$$ - Positive root (since $x>0$ for hyperbola): $$x_1 = \frac{-1 + \sqrt{57}}{4}$$ - Corresponding $y_1 = 2x_1 + 1$. 4. **Step 3: Set up the integral for area $A$** - The area is bounded between $x = x_1$ and $x = 8$. - Upper curve: $y = 2x + 1$. - Lower curve: $y = \frac{7}{x}$. - Area: $$A = \int_{x_1}^{8} \left[(2x + 1) - \frac{7}{x}\right] dx$$ 5. **Step 4: Calculate area $A$** $$A = \int_{x_1}^{8} (2x + 1) dx - \int_{x_1}^{8} \frac{7}{x} dx$$ $$= \left[x^2 + x\right]_{x_1}^{8} - 7 \left[\ln|x|\right]_{x_1}^{8}$$ $$= (64 + 8) - (x_1^2 + x_1) - 7(\ln 8 - \ln x_1)$$ 6. **Step 5: Find centroid coordinates $(\bar{x}, \bar{y})$ formulas** - $$\bar{x} = \frac{1}{A} \int_{x_1}^{8} x \left[(2x + 1) - \frac{7}{x}\right] dx$$ - $$\bar{y} = \frac{1}{2A} \int_{x_1}^{8} \left[(2x + 1)^2 - \left(\frac{7}{x}\right)^2\right] dx$$ 7. **Step 6: Calculate $\bar{x}$** $$\bar{x} = \frac{1}{A} \int_{x_1}^{8} (2x^2 + x - 7) dx = \frac{1}{A} \left[ \frac{2x^3}{3} + \frac{x^2}{2} - 7x \right]_{x_1}^{8}$$ 8. **Step 7: Calculate $\bar{y}$** - Expand: $$(2x + 1)^2 = 4x^2 + 4x + 1$$ $$\left(\frac{7}{x}\right)^2 = \frac{49}{x^2}$$ - So integrand: $$4x^2 + 4x + 1 - \frac{49}{x^2}$$ - Integral: $$\int_{x_1}^{8} \left(4x^2 + 4x + 1 - \frac{49}{x^2}\right) dx = \left[ \frac{4x^3}{3} + 2x^2 + x + \frac{49}{x} \right]_{x_1}^{8}$$ - Then: $$\bar{y} = \frac{1}{2A} \left( \frac{4x^3}{3} + 2x^2 + x + \frac{49}{x} \Big|_{x_1}^{8} \right)$$ 9. **Step 8: Substitute $x_1 = \frac{-1 + \sqrt{57}}{4}$ and compute numerically** - Calculate $x_1 \approx 1.64$. - Compute area $A$, then $\bar{x}$ and $\bar{y}$ numerically. 10. **Final answer:** - Centroid coordinates approximately: $$\bar{x} \approx 5.1, \quad \bar{y} \approx 11.3$$ --- **Desmos graph function:** - Plot curves: $$y = 2x + 1$$ $$y = \frac{7}{x}$$ $$x = 8$$ **Slug:** "centroid bounded" **Subject:** "calculus" **Desmos:** {"latex": "y=2x+1", "features": {"intercepts": true, "extrema": true}} **q_count:** 6