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Find the arclength of a parametric curve given below for 0 ≤ t ≤ (x(t) = 3 cost (y(t) = 3 sint

Find the arclength of a parametric curve given below for 0 ≤ t ≤ (x(t) = 3 cost (y(t) = 3 sint

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter8: Complex Numbers And Polarcoordinates
Section: Chapter Questions
Problem 8CLT
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**Title: Calculating the Arc Length of a Parametric Curve** **Description:** In this example, we will learn how to find the arc length of a parametric curve defined by the parametric equations \( x(t) \) and \( y(t) \) over the interval \( 0 \leq t \leq \pi \). **Problem Statement:** Find the arc length of the parametric curve given below for \( 0 \leq t \leq \pi \): \[ x(t) = 3 \cos t \] \[ y(t) = 3 \sin t \] **Explanation:** 1. **Parametric Equations:** - The x-component of the curve is given by \( x(t) = 3 \cos t \). - The y-component of the curve is given by \( y(t) = 3 \sin t \). 2. **Interval:** - The variable \( t \) ranges from \( 0 \) to \( \pi \). **Finding the Arc Length:** The arc length \( L \) of a parametric curve \((x(t), y(t))\) from \( t = a \) to \( t = b \) can be found using the formula: \[ L = \int_{a}^{b} \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 } \, dt \] For this specific problem: 1. **Compute the derivatives:** - \( \frac{dx}{dt} = -3 \sin t \) - \( \frac{dy}{dt} = 3 \cos t \) 2. **Set up the integral:** - \( L = \int_{0}^{\pi} \sqrt{ (-3 \sin t)^2 + (3 \cos t)^2 } \, dt \) 3. **Simplify the integrand:** - \( L = \int_{0}^{\pi} \sqrt{ 9 \sin^2 t + 9 \cos^2 t } \, dt \) - \( L = \int_{0}^{\pi} \sqrt{ 9 (\sin^2 t + \cos^2 t) } \, dt \) - Since \( \sin^2 t + \
Transcribed Image Text:**Title: Calculating the Arc Length of a Parametric Curve** **Description:** In this example, we will learn how to find the arc length of a parametric curve defined by the parametric equations \( x(t) \) and \( y(t) \) over the interval \( 0 \leq t \leq \pi \). **Problem Statement:** Find the arc length of the parametric curve given below for \( 0 \leq t \leq \pi \): \[ x(t) = 3 \cos t \] \[ y(t) = 3 \sin t \] **Explanation:** 1. **Parametric Equations:** - The x-component of the curve is given by \( x(t) = 3 \cos t \). - The y-component of the curve is given by \( y(t) = 3 \sin t \). 2. **Interval:** - The variable \( t \) ranges from \( 0 \) to \( \pi \). **Finding the Arc Length:** The arc length \( L \) of a parametric curve \((x(t), y(t))\) from \( t = a \) to \( t = b \) can be found using the formula: \[ L = \int_{a}^{b} \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 } \, dt \] For this specific problem: 1. **Compute the derivatives:** - \( \frac{dx}{dt} = -3 \sin t \) - \( \frac{dy}{dt} = 3 \cos t \) 2. **Set up the integral:** - \( L = \int_{0}^{\pi} \sqrt{ (-3 \sin t)^2 + (3 \cos t)^2 } \, dt \) 3. **Simplify the integrand:** - \( L = \int_{0}^{\pi} \sqrt{ 9 \sin^2 t + 9 \cos^2 t } \, dt \) - \( L = \int_{0}^{\pi} \sqrt{ 9 (\sin^2 t + \cos^2 t) } \, dt \) - Since \( \sin^2 t + \
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