Final answer:
The length of the curve given by r(t) = 2t i + et j + e(-t) k from t = 0 to t = 5 is found by integrating the magnitude of the derivative dr/dt from 0 to 5. The magnitude is |v(t)| = sqrt(4 + e^(2t) + e^(-2t)), and the integral is performed over the given interval to obtain the curve length.
Step-by-step explanation:
The student is asking for the length of the curve represented by the vector function r(t) = 2t i + et j + e(-t) k from t = 0 to t = 5. To find the length of a curve given by a vector function, you typically integrate the magnitude of the velocity vector of the curve over the given interval. The velocity vector is found by taking the derivative of the position vector r(t).
First, calculate the derivative dr/dt to find the velocity vector:
- The derivative of 2t i with respect to t is 2 i.
- The derivative of et j with respect to t is et j.
- The derivative of e(-t) k with respect to t is -e(-t) k.
So the velocity vector is v(t) = 2 i + et j - e(-t) k. The magnitude of this vector is |v(t)| = sqrt(22 + e2t + e-2t). To find the length of the curve L, integrate |v(t)| from 0 to 5.
Therefore, the curve's length L is given by the integral \(\int_{0}^{5} sqrt(4 + e^{2t} + e^{-2t}) dt\). This requires the use of certain integration techniques, possibly numerical integration, as it likely does not have a closed-form antiderivative.