Problem 3 - Calculus Early Transcendentals

Question

Suppose \(f\) is positive and differentiable on \([a, b] .\) The curve \(y=f(x)\) on \([a, b]\) is revolved about the \(x\) -axis. Explain how to find the area of the surface that is generated.

Step-by-Step Solution

Answer

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.

Step 1: Determine the derivative of the function

To begin, determine the derivative of the function, \(f'(x)\), and then square the value. The derivative represents the slope of the tangent line at any point on the curve \(y=f(x)\). The reason we square the derivative is that it will be needed for the surface area formula mentioned in the analysis section.

Step 2: Add 1 to the squared derivative

Next, add 1 to the squared derivative: \(1 + (f'(x))^2\). The reason we do this step is that this will be part of our formula for surface area.

Step 3: Calculate the square root of the sum

Calculate the square root of the sum obtained in step 2: \(\sqrt{1 + (f'(x))^2}\). This step is necessary as it also forms part of the surface area formula.

Step 4: Multiply by the function

Multiply the original function, \(f(x)\), by the result obtained in step 3: \(f(x) \sqrt{1 + (f'(x))^2}\). This result represents the integrand for the surface area of revolution.

Step 5: Find the surface area

Finally, use the formula for the surface area of revolution about the \(x\)-axis (as mentioned in the analysis section) by setting up the integral and multiplying by \(2 \pi\): \(S = 2\pi\int_a^b f(x)\sqrt{1 + (f'(x))^2} dx\). Evaluate this integral to find the surface area \(S\) that is generated when the curve \(y=f(x)\) on \([a, b]\) is revolved around the \(x\)-axis.