https://file.aiquickdraw.com/mathbot/file/1729041538605z6ciikr5.jpg

To find the length of the parametric curve given by:
\[
x = t \cos(t), \quad y = t - 7 \sin(t),
\]
we use the formula for the length of a parametric curve, which is:
\[
L = \int_{a}^{b} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt.
\]

### Step 1: Find \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \).
- Given \( x = t \cos(t) \), we compute:
  \[
  \frac{dx}{dt} = \cos(t) - t \sin(t).
  \]

- Given \( y = t - 7 \sin(t) \), we compute:
  \[
  \frac{dy}{dt} = 1 - 7 \cos(t).
  \]

### Step 2: Set up the integral.
The integral becomes:
\[
L = \int_{-\pi}^{\pi} \sqrt{(\cos(t) - t \sin(t))^2 + (1 - 7 \cos(t))^2} \, dt.
\]

### Step 3: Evaluate the integral using a calculator or computer.
The length of the curve can be found by calculating the above integral numerically.

Let me know if you want me to proceed with the numerical evaluation or if you have any other questions.

---

Would you like details on the calculation, or do you have any other questions?

Here are 5 related questions to explore:
1. How do you find the arc length of a parametric curve over different intervals?
2. How would the calculation change if \( x \) and \( y \) were functions of different variables?
3. What is the significance of the parameter range \( -\pi \) to \( \pi \) in this problem?
4. How does changing the parameter in the equation of \( y \) affect the curve's shape?
5. Can the length of this curve be found analytically, or must it be done numerically?

**Tip:** When dealing with parametric equations, always compute the derivatives accurately to avoid errors in calculating the arc length or other properties.

Set up an integral that represents the length of the part of the parametric curve shown in the graph, where x = t * cos(t) and y = t - 7 * sin(t).

Learn how to set up an integral to find the length of a parametric curve, where x = t * cos(t) and y = t - 7 * sin(t), from -π to π. Detailed steps include differentiation and setting up the arc length formula.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation