Final answer:
Calculating the length of a curve typically involves integrating the square root of 1 plus the square of the derivative of the function describing the curve. The exact methods vary based on the given limits and the specific function.
Step-by-step explanation:
To find the length of the curve S(x) = ∫ₐᵗ √(1 + eᵗ) dt, we need to know the limits of integration, which are not provided in the question. Assuming 'a' is the lower limit and 'x' is the upper limit, we would integrate the function within those limits to find S(x).
For the second request, which involves finding the length of the curve described by y = ln(cos x) within the domain 0 ≤ x ≤ π/4, we use the formula for the arc length of a curve in the form y = f(x):
L = ∫ from a to b √(1 + (f'(x))^2) dx.
To apply this, differentiate ln(cos x) with respect to x to get f'(x) = -tan(x), and then integrate the square root of 1 plus the square of this derivative from 0 to π/4. However, since the function ln(cos x) is not defined for x = π/2, this question might contain a typo or an incorrect domain, as ln(cos x) is not real for x > π/4.
For generalized curve length questions, the same arc length formula is used by finding the derivative of the function, squaring it, and then integrating with the relevant limits.