Find the length of the curve.
r(t) = 

sqrt2
 t i + et j + e−t k,   0 ≤ t ≤ 5

The length of a curve given by a vector function \(\mathbf{r}(t)\) from \(t = a\) to \(t = b\) is calculated using the formula:

\[
L = \int_a^b \|\mathbf{r}'(t)\| \, dt
\]

Given the vector function:

\[
\mathbf{r}(t) = \sqrt{2} \, t \, \mathbf{i} + e^t \, \mathbf{j} + e^{-t} \, \mathbf{k}
\]

1. **Step 1: Differentiate the components of \(\mathbf{r}(t)\)**
   
   The derivative \(\mathbf{r}'(t)\) is found by differentiating each component with respect to \(t\).

   \[
   \mathbf{r}'(t) = \frac{d}{dt} \left( \sqrt{2} t \right) \mathbf{i} + \frac{d}{dt} \left( e^t \right) \mathbf{j} + \frac{d}{dt} \left( e^{-t} \right) \mathbf{k}
   \]

   \[
   \mathbf{r}'(t) = \sqrt{2} \mathbf{i} + e^t \mathbf{j} - e^{-t} \mathbf{k}
   \]

2. **Step 2: Find the magnitude of \(\mathbf{r}'(t)\)**

   The magnitude of \(\mathbf{r}'(t)\) is:

   \[
   \|\mathbf{r}'(t)\| = \sqrt{(\sqrt{2})^2 + (e^t)^2 + (-e^{-t})^2}
   \]

   Simplifying:

   \[
   \|\mathbf{r}'(t)\| = \sqrt{2 + e^{2t} + e^{-2t}}
   \]

3. **Step 3: Set up the integral for the length**

   The length of the curve is given by:

   \[
   L = \int_0^5 \sqrt{2 + e^{2t} + e^{-2t}} \, dt
   \]

This integral may not have a simple closed-form solution, so it can be evaluated numerically.

I'll compute this integral for you.The length of the curve \(\mathbf{r}(t) = \sqrt{2} \, t \, \mathbf{i} + e^t \, \mathbf{j} + e^{-t} \, \mathbf{k}\) from \(t = 0\) to \(t = 5\) is approximately **148.41 units**.

Would you like further details or have any questions?

Here are 5 related questions to deepen understanding:
1. How do you interpret the physical meaning of the length of a curve in 3D space?
2. What is the significance of the parameter \(t\) in the curve equation?
3. How would you find the length if the limits of \(t\) were different?
4. What numerical methods are used for evaluating integrals that don't have closed-form solutions?
5. How do you compute arc length for curves in 2D versus 3D?

**Tip:** When dealing with vector functions, always calculate the derivative of each component separately before finding the magnitude of the vector.

Find the length of the curve r(t) = sqrt(2) t i + e^t j + e^(−t) k,   0 ≤ t ≤ 5

Discover how to calculate the length of a 3D curve r(t) = sqrt(2) t i + e^t j + e^(-t) k over the interval 0 ≤ t ≤ 5. This example includes the arc length formula, vector differentiation, and numerical integration methods. Suitable for university-level calculus students.

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