Final answer:
To determine the precise length of the curve y = 1 - e^(-x), the arc length formula is applied: L = ∫[0, x] sqrt(1 + e^(-2x)) dx. Simplifying and integrating, the exact length is given by L = x + sqrt(x^2 + 1) + ln(x + sqrt(x^2 + 1)) + C, where C is the constant of integration.
Step-by-step explanation:
To find the exact length of the curve y = 1 - e^(-x), we can use the arc length formula for a function y = f(x) on an interval [a, b]. The formula is given by:
L = ∫[a, b] sqrt(1 + (f'(x))^2) dx
In this case, f(x) = 1 - e^(-x), so f'(x) = e^(-x). Plugging this into the formula and integrating from 0 to x, we get:
L = ∫[0, x] sqrt(1 + (e^(-x))^2) dx
Simplifying the integrand and evaluating the integral, we find the exact length of the curve y = 1 - e^(-x) to be:
L = x + sqrt(x^2 + 1) + ln(x + sqrt(x^2 + 1)) + C