A surface is generated by revolving the line f(x)=2−x, for 0≤x≤2, about the x-axis. Find the area of the resulting surface in the following ways.


a. Using calculus

Verified step by step guidance

1. Step 1: Recall the formula for the surface area of a solid of revolution about the x-axis. The formula is: $A\; =\; 2\pi \; \int [a,b]\; f(x)\surd (1\; +\; (f\text{'}(x))²)\; dx$, where f(x) is the function being revolved, and f'(x) is its derivative.
2. Step 2: Identify the function f(x) = 2 - x and its domain [0, 2]. Compute the derivative of f(x): $f\text{'}(x)\; =\; -1$.
3. Step 3: Substitute f(x) and f'(x) into the formula. The integrand becomes: $2\pi \; \int [0,2]\; (2\; -\; x)\surd (1\; +\; (-1)²)\; dx$. Simplify the square root term: $\surd (1\; +\; 1)\; =\; \surd 2$, so the integrand is $2\pi \surd 2\; \int [0,2]\; (2\; -\; x)\; dx$.
4. Step 4: Break down the integral $\int [0,2]\; (2\; -\; x)\; dx$ into two simpler integrals: $\int [0,2]\; 2\; dx\; -\; \int [0,2]\; x\; dx$. Compute each integral separately using the power rule for integration.
5. Step 5: Multiply the result of the integral by $2\pi \surd 2$ to find the surface area. This completes the calculation process.

Key Concepts

Surface Area of Revolution

The surface area of a solid of revolution is calculated by revolving a curve around an axis. For a function f(x) revolved around the x-axis, the formula is given by A = 2π ∫[a to b] f(x) √(1 + (f'(x))^2) dx, where [a, b] is the interval of x. This formula accounts for both the height of the function and the slope, providing the total surface area generated.

Definite Integral

A definite integral represents the accumulation of quantities, such as area under a curve, over a specific interval [a, b]. It is denoted as ∫[a to b] f(x) dx and provides a numerical value that corresponds to the total area between the curve f(x) and the x-axis from x = a to x = b. Understanding how to evaluate definite integrals is crucial for calculating surface areas.

Derivative and Its Role

The derivative of a function, denoted as f'(x), measures the rate of change of the function with respect to x. In the context of surface area, the derivative is used to find the slope of the curve, which is essential for the formula involving √(1 + (f'(x))^2). This term adjusts the length of the curve segment being revolved, ensuring accurate surface area calculations.