To find the exact length of the curve defined by x = e^t + e^(-t) and y = 5 - 2t between t = 0 and t = 3, we can use the arc length formula:
L = ∫[a, b] √(dx/dt)^2 + (dy/dt)^2 dt
First, let's find the derivatives dx/dt and dy/dt:
dx/dt = e^t - e^(-t)
dy/dt = -2
Now, we can substitute these derivatives into the arc length formula and integrate:
L = ∫[0, 3] √((e^t - e^(-t))^2 + (-2)^2) dt
Simplifying the integrand:
L = ∫[0, 3] √(e^(2t) - 2 + e^(-2t) + 4) dt
Integrating this expression is quite complex and cannot be done exactly. However, it can be approximated using numerical methods or software.
Therefore, we cannot find the exact length of the curve, but we can approximate it using numerical methods.