Find length of the curve r(t) = ⟨2t, t2, 31 t3⟩ from t = 1 to t = 4

To find the length of the curve \( \mathbf{r}(t) = \langle 2t, t^2, 31t^3 \rangle \) from \( t = 1 \) to \( t = 4 \), we use the formula for the arc length of a vector-valued function:

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

### Step 1: Find the derivative \( \mathbf{r}'(t) \)
The derivative of the vector function \( \mathbf{r}(t) = \langle 2t, t^2, 31t^3 \rangle \) is:

\[
\mathbf{r}'(t) = \langle 2, 2t, 93t^2 \rangle
\]

### Step 2: Compute the magnitude of \( \mathbf{r}'(t) \)
Now, compute the magnitude of the derivative \( \mathbf{r}'(t) \):

\[
\left| \mathbf{r}'(t) \right| = \sqrt{(2)^2 + (2t)^2 + (93t^2)^2}
\]
\[
\left| \mathbf{r}'(t) \right| = \sqrt{4 + 4t^2 + 8649t^4}
\]

### Step 3: Set up the integral
The arc length \( L \) is given by:

\[
L = \int_{1}^{4} \sqrt{4 + 4t^2 + 8649t^4} \, dt
\]

This integral needs to be computed. Let's proceed with that calculation.

The length of the curve \( \mathbf{r}(t) = \langle 2t, t^2, 31t^3 \rangle \) from \( t = 1 \) to \( t = 4 \) is approximately \( 1953.08 \) units.

Do you want further details on this process, or do you have any other questions?

Here are some related questions:
1. How does the formula for arc length change in different coordinate systems?
2. How can you compute arc length if given a parametric curve in polar coordinates?
3. How does curvature relate to the arc length of a curve?
4. What are some real-world applications of arc length computations?
5. How does the arc length change if the function is given in higher dimensions?

**Tip:** When computing arc lengths, simplifying the integrand before solving can sometimes make the problem easier to handle manually.

Find the length of the curve r(t) = ⟨2t, t^2, 31t^3⟩ from t = 1 to t = 4.

Learn how to find the length of the curve r(t) = ⟨2t, t^2, 31t^3⟩ from t = 1 to t = 4 using arc length formulas in vector calculus. This problem covers key concepts like vector derivatives, integral calculus, and parametric curves. Suitable for students in undergraduate calculus.

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