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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help with cal hw
![**Problem Description:**
What is the length of the polar curve \( r = 7 \sin \theta \) from \( \theta = 0 \) to \( \theta = 2\pi \)?
*Note: Enter a numerical value only, rounding your answer to one decimal place.*
**Explanation:**
This problem asks for the length of a polar curve defined by the equation \( r = 7 \sin \theta \). The range of \(\theta\) is from 0 to \(2\pi\). This is a mathematical problem involving integration to calculate the arc length of the curve in polar coordinates. To solve this, one would use the formula for the arc length of a polar curve:
\[
L = \int_a^b \sqrt{\left(\frac{dr}{d\theta}\right)^2 + r^2} \, d\theta
\]
Where \( r = 7 \sin \theta \) and the limits of integration are \( a = 0 \) and \( b = 2\pi \).
To find the solution, you will need to:
1. Compute \(\frac{dr}{d\theta}\).
2. Substitute \( r \) and \(\frac{dr}{d\theta}\) into the arc length formula.
3. Evaluate the integral for \(\theta\) ranging from 0 to \(2\pi\).
4. Round the result to one decimal place.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F286060ef-b6f0-4e93-baf7-6d09e677932d%2F9fbbd772-927c-466c-8c02-0be0ce63fd80%2F43a1ceh_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Description:**
What is the length of the polar curve \( r = 7 \sin \theta \) from \( \theta = 0 \) to \( \theta = 2\pi \)?
*Note: Enter a numerical value only, rounding your answer to one decimal place.*
**Explanation:**
This problem asks for the length of a polar curve defined by the equation \( r = 7 \sin \theta \). The range of \(\theta\) is from 0 to \(2\pi\). This is a mathematical problem involving integration to calculate the arc length of the curve in polar coordinates. To solve this, one would use the formula for the arc length of a polar curve:
\[
L = \int_a^b \sqrt{\left(\frac{dr}{d\theta}\right)^2 + r^2} \, d\theta
\]
Where \( r = 7 \sin \theta \) and the limits of integration are \( a = 0 \) and \( b = 2\pi \).
To find the solution, you will need to:
1. Compute \(\frac{dr}{d\theta}\).
2. Substitute \( r \) and \(\frac{dr}{d\theta}\) into the arc length formula.
3. Evaluate the integral for \(\theta\) ranging from 0 to \(2\pi\).
4. Round the result to one decimal place.
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