Find the length of the curve <math xmlns="http://www.w3.org/1998/Math/MathML">
  <mstyle displaystyle="true">
    <mover>
      <mi>r</mi>
      <mo>&#xAF;</mo>
    </mover>
    <mrow>
      <mo>(</mo>
      <mi>t</mi>
      <mo>)</mo>
    </mrow>
    <mo>=</mo>
    <mrow>
      <mo>&#x2329;</mo>
      <mn>4</mn>
      <msup>
        <mi>t</mi>
        <mn>2</mn>
      </msup>
      <mo>,</mo>
      <mn>4</mn>
      <mi>t</mi>
      <mo>,</mo>
      <mrow>
        <mi>ln</mi>
        <mrow>
          <mo>(</mo>
          <mi>t</mi>
          <mo>)</mo>
        </mrow>
      </mrow>
      <mo>&#x232A;</mo>
    </mrow>
  </mstyle>
</math>
 
 for <math xmlns="http://www.w3.org/1998/Math/MathML">
  <mstyle displaystyle="true">
    <mn>1</mn>
    <mo>&#x2264;</mo>
    <mi>t</mi>
    <mo>&#x2264;</mo>
    <mn>5</mn>
  </mstyle>
</math>

To find the length of the curve \(\overline{r}(t)\) over the interval \(1 \leq t \leq 5\), we need to apply the formula for the arc length of a parametric curve:

\[
L = \int_{t_1}^{t_2} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 + \left( \frac{dz}{dt} \right)^2} \, dt
\]

In this case, the parametric curve is given by:

\[
\overline{r}(t) = \langle 4t^2, 4t, \ln(t) \rangle
\]

### Step 1: Compute the derivatives
First, we compute the derivatives of each component of \(\overline{r}(t)\):

- \( x(t) = 4t^2 \), so \(\frac{dx}{dt} = 8t\)
- \( y(t) = 4t \), so \(\frac{dy}{dt} = 4\)
- \( z(t) = \ln(t) \), so \(\frac{dz}{dt} = \frac{1}{t}\)

### Step 2: Substitute into the arc length formula
Now we substitute these derivatives into the formula for the arc length:

\[
L = \int_{1}^{5} \sqrt{(8t)^2 + (4)^2 + \left(\frac{1}{t}\right)^2} \, dt
\]

Simplifying inside the square root:

\[
L = \int_{1}^{5} \sqrt{64t^2 + 16 + \frac{1}{t^2}} \, dt
\]

### Step 3: Compute the integral
To find the exact value of the length of the curve, we would now compute the integral:

\[
L = \int_{1}^{5} \sqrt{64t^2 + 16 + \frac{1}{t^2}} \, dt
\]

This integral doesn't have a simple antiderivative, so it would be evaluated numerically. Let me compute this for you.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to proceed with a more detailed step-by-step explanation of how to set up the integral for numerical evaluation, or do you have any other questions?

Here are 5 related questions for further exploration:
1. How would you compute the arc length if one of the coordinates had a more complex function, like \(e^t\)?
2. What are some methods to approximate integrals like this one numerically?
3. Can the arc length formula be extended to curves in higher-dimensional spaces?
4. How does the arc length change when the curve is reparametrized?
5. What is the geometric interpretation of the arc length of a curve?

**Tip:** Numerical integration is often the go-to technique for arc length problems when the integral does not have a simple analytical solution.

Find the length of the curve r̅(t) = ⟨4t², 4t, ln(t)⟩ for 1 ≤ t ≤ 5.

Learn how to find the arc length of the parametric curve r(t) = ⟨4t², 4t, ln(t)⟩ for t in the interval [1, 5]. This problem involves key calculus concepts such as parametric curves, derivatives, and the arc length formula. Suitable for undergraduate calculus 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