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: Calculating Arc Length**
Find the arc length of the curve defined by the parametric equations:
\[ x = \frac{1}{3} t^3, \quad y = \frac{1}{2} t^2 \]
where the parameter \( t \) ranges from 2 to 7.
**Steps for Solution:**
To find the arc length of a parametric curve \( (x(t), y(t)) \) from \( t = a \) to \( t = b \), use the arc length formula:
\[ L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \]
1. Differentiate \( x(t) = \frac{1}{3}t^3 \) with respect to \( t \) to get \( \frac{dx}{dt} \).
2. Differentiate \( y(t) = \frac{1}{2}t^2 \) with respect to \( t \) to get \( \frac{dy}{dt} \).
3. Substitute \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) into the arc length formula.
4. Evaluate the integral from \( t = 2 \) to \( t = 7 \) to find the arc length.
Proceed with these steps to solve the problem mathematically.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd21e586c-08df-43d0-b266-231fca0b584a%2Ff5b9ecfd-e963-457b-920c-ebbebe15e146%2Fx9n8rpg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem: Calculating Arc Length**
Find the arc length of the curve defined by the parametric equations:
\[ x = \frac{1}{3} t^3, \quad y = \frac{1}{2} t^2 \]
where the parameter \( t \) ranges from 2 to 7.
**Steps for Solution:**
To find the arc length of a parametric curve \( (x(t), y(t)) \) from \( t = a \) to \( t = b \), use the arc length formula:
\[ L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \]
1. Differentiate \( x(t) = \frac{1}{3}t^3 \) with respect to \( t \) to get \( \frac{dx}{dt} \).
2. Differentiate \( y(t) = \frac{1}{2}t^2 \) with respect to \( t \) to get \( \frac{dy}{dt} \).
3. Substitute \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) into the arc length formula.
4. Evaluate the integral from \( t = 2 \) to \( t = 7 \) to find the arc length.
Proceed with these steps to solve the problem mathematically.
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