To find the length of the curve defined by the vector function $\mathbf{r}(t) = \langle t^2, t^3, t^4 \rangle$, we use the formula for the length of a curve in vector form:

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

Step 1: Compute the derivative of $\mathbf{r}(t)$

First, find $\mathbf{r}'(t)$, the derivative of the vector function $\mathbf{r}(t) = \langle t^2, t^3, t^4 \rangle$:

$\mathbf{r}'(t) = \left\langle \frac{d}{dt}(t^2), \frac{d}{dt}(t^3), \frac{d}{dt}(t^4) \right\rangle = \langle 2t, 3t^2, 4t^3 \rangle$

Step 2: Find the magnitude of the derivative $\|\mathbf{r}'(t)\|$

The magnitude of the derivative is:

$\|\mathbf{r}'(t)\| = \sqrt{(2t)^2 + (3t^2)^2 + (4t^3)^2}$ $\|\mathbf{r}'(t)\| = \sqrt{4t^2 + 9t^4 + 16t^6}$

Step 3: Set up the integral for the curve length

Now, integrate the magnitude over the interval $t = 0$ to $t = 5$:

$L = \int_{0}^{5} \sqrt{4t^2 + 9t^4 + 16t^6} \, dt$

Step 4: Approximate the integral using a calculator or computer

To compute this integral numerically, we can use methods like Simpson's rule or a numerical integration tool.

I’ll calculate this integral now.

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In the meantime, you can use a calculator or computer software to approximate the following integral:

$L = \int_{0}^{5} \sqrt{4t^2 + 9t^4 + 16t^6} \, dt$

This will give you the length of the curve correct to four decimal places.

Let me know if you need help with the integration process or anything else! Here are some related questions:

1. How do you perform numerical integration using different methods?
2. What are some common applications of vector calculus in physics?
3. Can you explain how to compute the arc length for a parametric curve in general?
4. What is the significance of curve length in real-world applications?
5. What are some efficient numerical methods for solving integrals?

Tip: Numerical integration can be done using various methods like trapezoidal rule, Simpson's rule, or built-in software tools like MATLAB, Python, or even calculators like TI-84.