**Problem Statement:**Find the length of the curve defined by the parametric equations:\[ x = \cos t \]\[ y = \sin t \]for the parameter interval \( 0 \leq t \leq 2\pi \).**Explanation:**The equations represent the parametric form of a circle with a radius of 1. The task is to calculate the circumference of this circle, given the range of the parameter \( t \) from 0 to \( 2\pi \).**Calculation Steps:**1. **Identify the Curve:** - The equations describe a unit circle centered at the origin in the Cartesian plane.2. **Use the Formula for Arc Length of a Parametric Curve:** - The arc length \( L \) of a parametric curve from \( t = a \) to \( t = b \) is given by: \[ L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \]3. **Compute Derivatives:** - \( \frac{dx}{dt} = -\sin t \) - \( \frac{dy}{dt} = \cos t \)4. **Substitute in the Arc Length Formula:** - \( \left(\frac{dx}{dt}\right)^2 = \sin^2 t \) - \( \left(\frac{dy}{dt}\right)^2 = \cos^2 t \) - Therefore, \( \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 = \sin^2 t + \cos^2 t = 1 \)5. **Simplify and Calculate:** - The integral becomes: \[ L = \int_{0}^{2\pi} \sqrt{1} \, dt = \int_{0}^{2\pi} 1 \, dt = [t]_{0}^{2\pi} = 2\pi \]**Conclusion:**The length of the curve, which is the circumference of the unit circle, is \( 2\pi \).

Name: **Problem Statement:**Find the length of the curve defined by the par
Uploaded: 2024-03-20T06:46:42.000Z
Channel: Bartleby
Description: **Problem Statement:**Find the length of the curve defined by the parametric equations:\[ x = \cos t \]\[ y = \sin t \]for the parameter interval \( 0 \leq t \leq 2\pi \).**Explanation:**The equations represent the parametric form of a circle with a