Final answer:
The arc length of a curve can be calculated using the integral of the derivative of the curve's equation. In this case, we need to find the derivative of the curve and then integrate it over the given range.
Step-by-step explanation:
The arc length of a curve is calculated using the formula:
Arc length = ∫r'(t)dt
Given the equation of the curve r(t) = <2√(2t), e^(2t), e^(-2t)>, we need to find the derivative of r(t).
Differentiating each component of r(t) with respect to t, we get:
r'(t) = <√(2/t), 2e^(2t), -2e^(-2t)>
Substituting the derivative r'(t) into the arc length formula, we have:
Arc length = ∫|r'(t)|dt
Integrating the magnitude of r'(t) over the given range will give us the arc length of the curve.