Find the arclength of the curve z = 3 cos (7t), y = 3 sin(7t) with 0 < t < -cot (71) Question Help: Message instructor Add Work x syntax incomplete.
Find the arclength of the curve z = 3 cos (7t), y = 3 sin(7t) with 0 < t < -cot (71) Question Help: Message instructor Add Work x syntax incomplete.
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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![### Problem Statement
Find the arc length of the curve \( x = 3 \cos(7t), y = 3 \sin(7t) \) with \(0 \leq t \leq \frac{\pi}{28}\).
### Submission
- \(-\cot(7t)\)
**Status**: Incorrect
**Comment**: Syntax incomplete.
### Assistance
- **Question Help**: Message instructor
- **Add Work**: Option available
---
### Explanation
The problem requires finding the arc length of a parametric curve defined by the equations \( x = 3 \cos(7t) \) and \( y = 3 \sin(7t) \) within the given range for the parameter \( t \).
#### To find the arc length of a parametric curve:
Use the formula for the arc length \( L \) of a parametric curve \((x(t), y(t))\) from \( t = a \) to \( t = b \):
\[ L = \int_{a}^{b} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt \]
Given:
- \( x = 3 \cos(7t) \)
- \( y = 3 \sin(7t) \)
- \( 0 \leq t \leq \frac{\pi}{28} \)
#### Steps to solve:
1. Find \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \):
\[
\frac{dx}{dt} = \frac{d}{dt} (3 \cos(7t)) = 3 \cdot (-7 \sin(7t)) = -21 \sin(7t)
\]
\[
\frac{dy}{dt} = \frac{d}{dt} (3 \sin(7t)) = 3 \cdot (7 \cos(7t)) = 21 \cos(7t)
\]
2. Substitute \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) into the arc length formula:
\[
L = \int_{0}^{\frac{\pi}{28}} \sqrt{(-21 \sin](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F41a4f8e2-50b1-40c5-ab3a-6358b729bd6f%2F4b71b192-6a30-4525-bb78-100e3780a054%2Fa2y67d9_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Find the arc length of the curve \( x = 3 \cos(7t), y = 3 \sin(7t) \) with \(0 \leq t \leq \frac{\pi}{28}\).
### Submission
- \(-\cot(7t)\)
**Status**: Incorrect
**Comment**: Syntax incomplete.
### Assistance
- **Question Help**: Message instructor
- **Add Work**: Option available
---
### Explanation
The problem requires finding the arc length of a parametric curve defined by the equations \( x = 3 \cos(7t) \) and \( y = 3 \sin(7t) \) within the given range for the parameter \( t \).
#### To find the arc length of a parametric curve:
Use the formula for the arc length \( L \) of a parametric curve \((x(t), y(t))\) from \( t = a \) to \( t = b \):
\[ L = \int_{a}^{b} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt \]
Given:
- \( x = 3 \cos(7t) \)
- \( y = 3 \sin(7t) \)
- \( 0 \leq t \leq \frac{\pi}{28} \)
#### Steps to solve:
1. Find \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \):
\[
\frac{dx}{dt} = \frac{d}{dt} (3 \cos(7t)) = 3 \cdot (-7 \sin(7t)) = -21 \sin(7t)
\]
\[
\frac{dy}{dt} = \frac{d}{dt} (3 \sin(7t)) = 3 \cdot (7 \cos(7t)) = 21 \cos(7t)
\]
2. Substitute \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) into the arc length formula:
\[
L = \int_{0}^{\frac{\pi}{28}} \sqrt{(-21 \sin
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