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find the length of the given curve over the given interval

To find the length of the curve given by the vector functions \( \mathbf{R}(t) \) over the specified intervals, we can use the formula for the arc length of a vector-valued function:

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

where \( \mathbf{R}(t) \) is the vector function and \( \mathbf{R}'(t) \) is its derivative. Now, let's calculate the length for each function.

### Problem 13:
The vector function is:

\[
\mathbf{R}(t) = (\cos^3 t) \mathbf{i} + (\cos^2 t) \mathbf{k}
\]

We need to:
1. Compute the derivative \( \mathbf{R}'(t) \).
2. Find \( \|\mathbf{R}'(t)\| \), the magnitude of the derivative.
3. Integrate \( \|\mathbf{R}'(t)\| \) over \( t \in [0, \frac{\pi}{2}] \).

### Problem 14:
The vector function is:

\[
\mathbf{R}(t) = t\mathbf{i} + 2t\mathbf{j} + 3t\mathbf{k}
\]

We will follow the same steps as above, but integrate over \( t \in [0, 2] \).

Let me compute these values for both problems.The arc lengths for the two problems are as follows:

### Problem 13:
The length of the curve is given by the integral:

\[
L = \int_0^{\frac{\pi}{2}} \sqrt{9 \cos^2(t) + 4} \cdot |\sin(t)| \cdot |\cos(t)| \, dt
\]

This integral requires further evaluation using numerical methods or advanced techniques to solve.

### Problem 14:
The length of the curve is:

\[
L = 2\sqrt{14}
\]

This simplifies to approximately \( L \approx 7.48 \).

Would you like more details on the evaluation of the first integral, or do you have any questions?

Here are some related questions to explore further:
1. How do you compute the magnitude of a vector-valued function's derivative?
2. What are the conditions under which the formula for arc length applies?
3. How do trigonometric identities simplify integrals like the one in Problem 13?
4. How does changing the parameterization of a curve affect its arc length?
5. What are some applications of finding the arc length of curves in physics or engineering?

**Tip:** When calculating arc lengths of vector functions, make sure the curve is continuously differentiable over the interval of interest.

Find the length of the given curve over the given interval.

Learn how to find the arc length of vector-valued functions over specific intervals. This problem involves vector calculus concepts like the arc length formula and vector derivatives. Suitable for students in an undergraduate vector calculus course.

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