https://file.aiquickdraw.com/mathbot/file/1722924952973gzysmzhw.jpg

To set up the integral that computes the length of the path parameterized by \( \mathbf{c}(t) = (5t, t^2) \) for \( 0 < t < 2 \), follow these steps:

1. **Parameterize the curve:** The given curve is parameterized by \( \mathbf{c}(t) = (5t, t^2) \).

2. **Compute the derivatives:** Find the derivatives of each component with respect to \( t \).

   \[
   \mathbf{c}'(t) = \left(\frac{d}{dt}(5t), \frac{d}{dt}(t^2)\right) = (5, 2t)
   \]

3. **Compute the magnitude of the derivative vector:** The magnitude (or norm) of \( \mathbf{c}'(t) \) is given by:

   \[
   \|\mathbf{c}'(t)\| = \sqrt{(5)^2 + (2t)^2} = \sqrt{25 + 4t^2}
   \]

4. **Set up the integral:** The length of the path from \( t = 0 \) to \( t = 2 \) is given by the integral of the magnitude of the derivative vector over this interval.

   \[
   L = \int_{0}^{2} \|\mathbf{c}'(t)\| \, dt = \int_{0}^{2} \sqrt{25 + 4t^2} \, dt
   \]

So, the integral that computes the length of the path is:

\[
\int_{0}^{2} \sqrt{25 + 4t^2} \, dt
\]

I have set up the integral as requested, without evaluating it.

Do you want more details or have any questions?

Here are 8 related questions to expand your understanding:

1. How would you evaluate the integral \( \int_{0}^{2} \sqrt{25 + 4t^2} \, dt \)?
2. What is the geometric interpretation of the parameterized path \( \mathbf{c}(t) = (5t, t^2) \)?
3. How do you determine if a parameterized curve is smooth?
4. Can you generalize the formula for the length of a curve parameterized by \( \mathbf{c}(t) \)?
5. What would the integral look like if the parameterized path was \( \mathbf{c}(t) = (t^3, t^2) \)?
6. How does the length of the curve change if the interval changes, for example from \( t = 0 \) to \( t = 3 \)?
7. What are the applications of computing the length of a parameterized path in physics and engineering?
8. How would you parameterize a circle and compute its circumference using an integral?

**Tip:** When setting up integrals for the length of a path, always make sure to correctly compute the magnitude of the derivative vector. This ensures you capture the true geometric length of the curve.

Explore how to set up and understand the integral \( \int_{0}^{2} \sqrt{25 + 4t^2} \, dt \) to compute the length of a path parameterized by \( \mathbf{c}(t) = (5t, t^2) \). This problem involves calculus concepts such as parameterization and integral calculus, suitable for advanced high school and college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation