Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Find the length of the curve.**
\[ x = \cos t \quad y = \sin t \quad 0 \leq t \leq 2\pi \]
**Again, use the "Insert --> Equation" path to type your equations.**
**Show your work using the equation editor and explain your steps in words.**
---
This task involves finding the arc length of a parametric curve. You are given the parametric equations \( x = \cos t \) and \( y = \sin t \) within the interval \( 0 \leq t \leq 2\pi \). This represents a full circle in the unit circle in the Cartesian plane.
To find the length of the curve, apply the formula for arc length of a parametric curve:
\[
L = \int_{a}^{b} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt
\]
Substitute \( dx/dt = -\sin t \) and \( dy/dt = \cos t \) into the formula, and evaluate the integral over the given interval. It is important to show each step of your calculation and explain the integration process in detail.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8f32ed55-1a30-4945-85d1-41ddf9f2fcd9%2F2aaf9a1c-5929-49cf-87fa-18d3e49df51e%2F97w2acd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Find the length of the curve.**
\[ x = \cos t \quad y = \sin t \quad 0 \leq t \leq 2\pi \]
**Again, use the "Insert --> Equation" path to type your equations.**
**Show your work using the equation editor and explain your steps in words.**
---
This task involves finding the arc length of a parametric curve. You are given the parametric equations \( x = \cos t \) and \( y = \sin t \) within the interval \( 0 \leq t \leq 2\pi \). This represents a full circle in the unit circle in the Cartesian plane.
To find the length of the curve, apply the formula for arc length of a parametric curve:
\[
L = \int_{a}^{b} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt
\]
Substitute \( dx/dt = -\sin t \) and \( dy/dt = \cos t \) into the formula, and evaluate the integral over the given interval. It is important to show each step of your calculation and explain the integration process in detail.
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