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We are tasked with finding the length \( L \) of the curve described by the vector function:

\[
\mathbf{r}(t) = \mathbf{i} + 6t^2 \mathbf{j} + t^3 \mathbf{k}, \quad 0 \leq t \leq \sqrt{33}
\]

The formula for the length of a curve given by a vector function \( \mathbf{r}(t) \) is:

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

Here, \( \mathbf{r}'(t) \) represents the derivative of \( \mathbf{r}(t) \) with respect to \( t \), and \( \|\mathbf{r}'(t)\| \) is the magnitude of this derivative.

### Step 1: Compute \( \mathbf{r}'(t) \)
We differentiate each component of \( \mathbf{r}(t) = \mathbf{i} + 6t^2 \mathbf{j} + t^3 \mathbf{k} \):

\[
\mathbf{r}'(t) = \frac{d}{dt}(\mathbf{i} + 6t^2 \mathbf{j} + t^3 \mathbf{k}) = 0 \mathbf{i} + 12t \mathbf{j} + 3t^2 \mathbf{k}
\]

Thus, the derivative is:

\[
\mathbf{r}'(t) = 12t \mathbf{j} + 3t^2 \mathbf{k}
\]

### Step 2: Compute the magnitude \( \|\mathbf{r}'(t)\| \)
The magnitude of \( \mathbf{r}'(t) \) is:

\[
\|\mathbf{r}'(t)\| = \sqrt{(12t)^2 + (3t^2)^2}
\]

Simplifying:

\[
\|\mathbf{r}'(t)\| = \sqrt{144t^2 + 9t^4} = \sqrt{9t^2(16 + t^2)} = 3t\sqrt{16 + t^2}
\]

### Step 3: Set up the integral
Now, we can set up the integral for the curve length:

\[
L = \int_0^{\sqrt{33}} 3t\sqrt{16 + t^2} \, dt
\]

### Step 4: Solve the integral
We'll now proceed with solving this integral. Let me calculate that.The length of the curve is \( L = 279 \).

Would you like more details or have any questions about the solution?

Here are 5 related questions to consider:
1. How is the formula for the arc length of a curve derived from the vector function?
2. What are the steps for finding the magnitude of a derivative in parametric curves?
3. How would the arc length change if the limits of integration were different?
4. What if one of the components of the curve \( \mathbf{r}(t) \) had a trigonometric function?
5. How do we interpret the physical meaning of curve length in different contexts?

**Tip**: When finding arc length, always check if simplifying the integrand helps before applying more complex integration techniques.

Find the length of the curve r(t) = i + 6t^2 j + t^3 k, 0 ≤ t ≤ sqrt(33).

Learn how to calculate the length of the curve defined by the parametric equation r(t) = i + 6t^2 j + t^3 k for t in [0, sqrt(33)]. This problem involves vector calculus, parametric equations, and the arc length formula. Suitable for undergraduate mathematics students.

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