Final answer:
To find the exact area of the surface obtained by rotating the curve y = 5 - x about the x-axis, you can use the formula for the surface area of revolution. Evaluating the integral obtained from the formula will give you the exact area.
Step-by-step explanation:
To find the area of the surface obtained by rotating the curve y = 5 - x about the x-axis, we can use the formula for the surface area of revolution. The formula is given by A = ∫2πy√(1+(dy/dx)²)dx. First, we need to find the derivative of y with respect to x, which is dy/dx = -1. Substituting these values into the formula, we get A = ∫2π(5-x)√(1+1²)dx.
Simplifying the expression, we have A = ∫2π(5-x)√2dx. Integrating with respect to x, we get A = 2π∫(5-x)√2dx. Integrating further, we have A = 2π(5x - x²/2)√2. Evaluating the integral from 3 to 5, we get A = 2π[(5(5) - (5)²/2)√2 - (5(3) - (3)²/2)√2]. Simplifying this expression gives us the exact area of the surface obtained by rotating the curve about the x-axis.
After evaluating this expression, we get the exact area of the surface obtained by rotating the curve as 30π√2 square units.