Let's solve the problem step by step: **Given:** Curve \( y = 8\sqrt{x} \) for \( 1.2 \leq x \leq 3.4 \). We are to find the **surface area of revolution** about the **x-axis**. --- ### **Step 1: Formula for Surface Area of Revolution** For a curve \( y = f(x) \) revolved about the x-axis from \( x = a \) to \( x = b \): \[ S = \int_a^b 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx \] --- ### **Step 2: Find \(\frac{dy}{dx}\)** for \( y = 8\sqrt{x} \) - \( y = 8x^{1/2} \) - \( \frac{dy}{dx} = 8 \cdot \frac{1}{2}x^{-1/2} = 4x^{-1/2} = \frac{4}{\sqrt{x}} \) --- ### **Step 3: Plug into the Formula** \[ S = \int_{1.2}^{3.4} 2\pi (8\sqrt{x}) \sqrt{1 + \left(\frac{4}{\sqrt{x}}\right)^2} dx \] --- ### **Step 4: Simplify the Expression** \[ \left(\frac{4}{\sqrt{x}}\right)^2 = \frac{16}{x} \] \[ \sqrt{1 + \left(\frac{4}{\sqrt{x}}\right)^2} = \sqrt{1 + \frac{16}{x}} \] So the integrand becomes: \[ 2\pi \cdot 8\sqrt{x} \cdot \sqrt{1 + \frac{16}{x}} \] \[ = 16\pi \sqrt{x} \sqrt{1 + \frac{16}{x}} \] Combine under one square root: \[ \sqrt{x} \sqrt{1 + \frac{16}{x}} = \sqrt{x \left(1 + \frac{16}{x}\right)} = \sqrt{x + 16} \] So the integrand is: \[ 16\pi \sqrt{x + 16} \] --- ### **Step 5: Write the Final Integral** \[ \boxed{ S = \int_{1.2}^{3.4} 16\pi \sqrt{x + 16} \, dx } \] --- ### **Step 6: Integrate** Let’s integrate \( 16\pi \sqrt{x + 16} \): Let \( u = x + 16 \), so \( du = dx \). When \( x = 1.2 \), \( u = 17.2 \). When \( x = 3.4 \), \( u = 19.4 \). The integral becomes: \[ 16\pi \int_{17.2}^{19.4} u^{1/2} du = 16\pi \left[ \frac{2}{3} u^{3/2} \right]_{17.2}^{19.4} = \frac{32\pi}{3} \left[ (19.4)^{3/2} - (17.2)^{3/2} \right] \] --- ### **Step 7: Final Answer** \[ \boxed{ S = \frac{32\pi}{3} \left[ (19.4)^{3/2} - (17.2)^{3/2} \right] } \] --- **You can leave it in this form, or plug in the numbers for a decimal value if needed!** --- #### **Summary Table** | Step | Explanation | Result | |------|-------------|--------| | 1 | Surface area formula | \( S = \int_a^b 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx \) | | 2 | Differentiate \( y \) | \( \frac{dy}{dx} = \frac{4}{\sqrt{x}} \) | | 3 | Plug in | \( S = \int_{1.2}^{3.4} 16\pi \sqrt{x + 16} dx \) | | 4 | Substitute \( u = x + 16 \) | \( S = 16\pi \int_{17.2}^{19.4} u^{1/2} du \) | | 5 | Integrate | \( S = \frac{32\pi}{3} \left[ (19.4)^{3/2} - (17.2)^{3/2} \right] \) | --- **That’s the detailed solution!**