Final answer:
To find the length of the provided curve r(t), one must compute the integral of the magnitude of the curve's derivative over the interval [0, 6]. After calculating the velocity vector by differentiating each component of r(t), integrate its magnitude to determine the curve's total length.
Step-by-step explanation:
Finding the Length of the Curve
To find the length of the given curve r(t)=2√t i+e⁻ᵗ j + e⁻ᵗ k, over the interval 0 ≤ t ≤ 6, we need to compute the integral of the magnitude of the curve's derivative. The first step is to find the derivative of r(t) with respect to t, denoted as r'(t).
r'(t) gives us the velocity vector whose magnitude gives the speed function. The arc length s can then be found by integrating the speed function over the given interval.
To obtain r'(t), we differentiate each component of r(t):
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- The derivative of 2√t with respect to t is 1/√t.
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- The derivative of e⁻ᵗ with respect to t is −e⁻ᵗ.
Therefore, r'(t) = (1/√t)i - e⁻ᵗj - e⁻ᵗk. Now we can calculate the magnitude of r'(t): √( (1/√t)^2 + (-e⁻ᵗ)^2 + (-e⁻ᵗ)^2 ).
Integrate this magnitude from t=0 to t=6 to find the arc length s. Note that caution must be taken at t=0, as the function is not defined at this point. Proper limits or an alternate approach, such as L'Hopital's rule, may be necessary to resolve this issue.
Once the integral is computed responsibly, the result will describe the total length of the curve between the specified t values.