Final answer:
To find the length of the curve, we can use the formula for arc length.
Step-by-step explanation:
Arc length is the distance along a curved line or the boundary of a sector of a circle. It is the measure of the actual length of a part of a curve, as opposed to the straight line distance between its endpoints. The formula for calculating the arc length depends on the context, whether it's a portion of a circle or a more general curve.
To find the length of the curve, we can use the formula for arc length. The formula for arc length is given by:
L = ∫√(dx/dt)² + (dy/dt)² + (dz/dt)² dt
For the given curve r(t) = 2t i + et j + e⁽⁻ᵗ⁾ k, we can find the derivatives of x, y, and z with respect to t and substitute it into the formula to evaluate the integral. After integrating, we will get the length of the curve over the interval 0 ≤ t ≤ 5.