Final answer:
The curve length L is calculated using the arc length formula, which entails finding the derivatives of the parametric equations and integrating over the given interval.
Step-by-step explanation:
The student asked how to compute the length of the curve given by the parametric equations (2t, ∫ t, t²) over the interval 1 ≤ t ≤ 5. To find the curve length, L, we'll use the formula:
L = ∫ab √(dx/dt)² + (dy/dt)² + (dz/dt)² dt
Where (x(t), y(t), z(t)) = (2t, ∫ nt, t²), and we're given the interval from t = 1 to t = 5.
First, we need to differentiate x(t), y(t), z(t) with respect to t to find dx/dt, dy/dt, and dz/dt:
- dx/dt = d(2t)/dt = 2
- dy/dt = d(∫ nt)/dt = n
- dz/dt = d(t²)/dt = 2t
Next, we substitute these into the arc length formula and evaluate the integral:
L = ∫15 √(2)² + (n)² + (2t)² dt
However, the student provided 'Int' as part of the curve, which seems to be a typo or incorrectly transcribed. Assuming that 'nt' represents a constant multiplied by t, the integration cannot be completed without the proper function or constant value 'n'. Therefore, I must clarify this before proceeding with the calculation.