Final answer:
To find the exact area of the surface obtained by rotating the curve y = 1 ex about the x-axis for 0 ≤ x ≤ 1, we can use the formula for finding the area of a surface of revolution. The formula is given by A = 2π ∫(f(x)√(1+(f'(x))^2)) dx.
Step-by-step explanation:
To find the exact area of the surface obtained by rotating the curve y = 1 ex about the x-axis for 0 ≤ x ≤ 1, we can use the formula for finding the area of a surface of revolution. The formula is given by A = 2π ∫(f(x)√(1+(f'(x))^2)) dx, where f(x) is the function being rotated and f'(x) is its derivative.
In this case, f(x) = 1 ex. Taking the derivative of f(x) gives f'(x) = ex. Substituting these values into the formula, we have A = 2π ∫(1 ex √(1+(ex)^2)) dx. Evaluating this integral from 0 to 1 will give us the exact area of the surface.