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:**
Compute the arc length of the parameterized curve given by:
\[ x = 3t\sin(t) \]
\[ y = 3t\cos(t) \]
for the interval \( 0 \leq t \leq \frac{\pi}{4} \).
**Solution Approach:**
To find the arc length of a parameterized curve \((x(t), y(t))\) over an interval \([a, b]\), use the formula:
\[ L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \]
1. **Calculate the derivatives:**
- Find \( \frac{dx}{dt} \)
- Find \( \frac{dy}{dt} \)
2. **Substitute into the arc length formula:**
- Compute \( \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 \)
- Integrate the expression from \( 0 \) to \( \frac{\pi}{4} \)
This calculation provides the length of the given curve within the specified interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7949c58c-a906-41a4-97ee-5d0bfe78cf99%2F3d15a200-5bc6-4db3-9afc-39907e57b267%2Fqta577j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Compute the arc length of the parameterized curve given by:
\[ x = 3t\sin(t) \]
\[ y = 3t\cos(t) \]
for the interval \( 0 \leq t \leq \frac{\pi}{4} \).
**Solution Approach:**
To find the arc length of a parameterized curve \((x(t), y(t))\) over an interval \([a, b]\), use the formula:
\[ L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \]
1. **Calculate the derivatives:**
- Find \( \frac{dx}{dt} \)
- Find \( \frac{dy}{dt} \)
2. **Substitute into the arc length formula:**
- Compute \( \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 \)
- Integrate the expression from \( 0 \) to \( \frac{\pi}{4} \)
This calculation provides the length of the given curve within the specified interval.
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