91Ó°ÊÓ

Question: Explain how to find the area of the surface generated when the curve \(y=f(x)\) on \([a, b]\) is revolved around the \(x\)-axis. Answer: To find the area of the surface generated, follow these steps: 1. Determine the derivative of the function \(f'(x)\) and square it. 2. Add 1 to the squared derivative: \(1 + (f'(x))^2\). 3. Calculate the square root of the sum: \(\sqrt{1 + (f'(x))^2}\). 4. Multiply the original function, \(f(x)\), by the result from step 3: \(f(x) \sqrt{1 + (f'(x))^2}\). 5. Use the formula for the surface area of revolution about the \(x\)-axis: \(S = 2\pi\int_a^b f(x)\sqrt{1 + (f'(x))^2} dx\) and evaluate the integral to find the surface area. By following these steps, you will be able to determine the area of the surface generated when the curve \(y=f(x)\) on \([a, b]\) is revolved around the \(x\)-axis.