Final answer:
The length of the curve given by the parametric equations is found by integrating the square root of the sum of the squares of the derivatives of x and y with respect to t over the interval from t = 5 to t = 7.
Step-by-step explanation:
To find the length of the curve defined by the parametric equations x = (2/4)t and y = 2 ln((t/4)^2 - 1) from t = 5 to t = 7, we must integrate the square root of the sum of the squares of dx/dt and dy/dt over the interval from 5 to 7.
First, we differentiate both x and y with respect to t:
- dx/dt = 2/4
- dy/dt = 2 × (1/2)t^{-1} × 2(t/4)
Then, we integrate the square root of the sum of these derivatives squared:
∫57 √{(dx/dt)^2 + (dy/dt)^2} dt
This integral calculates the arc length of the curve between the specified parametric bounds.