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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![**Example Problem: Finding the Length of a Polar Curve**
In this example, we are tasked with finding the exact length of a polar curve defined by the function \( r = e^{8\theta} \), where \( 0 \leq \theta \leq 2\pi \).
### Problem Statement:
**Find the exact length of the polar curve.**
\[ r = e^{8\theta}, \quad 0 \leq \theta \leq 2\pi \]
There is a box below the problem statement where the calculated numeric answer can be written.
### Solution Approach:
To find the length of a polar curve, you can use the formula for the arc length \( L \) of a curve given in polar coordinates:
\[ L = \int_{\alpha}^{\beta} \sqrt{\left( \frac{dr}{d\theta} \right)^2 + r^2} \, d\theta \]
Here:
- \( r = e^{8\theta} \)
- \( \alpha = 0 \)
- \( \beta = 2\pi \)
We need to take the following steps:
1. Calculate \( \frac{dr}{d\theta} \).
2. Substitute \( r \) and \( \frac{dr}{d\theta} \) into the arc length formula.
3. Evaluate the integral from \( \theta = 0 \) to \( \theta = 2\pi \).
### Details:
1. Calculate \( \frac{dr}{d\theta} \):
\[ \frac{dr}{d\theta} = \frac{d}{d\theta} (e^{8\theta}) = 8e^{8\theta} \]
2. Substitute into the arc length formula:
\[ L = \int_{0}^{2\pi} \sqrt{\left( 8e^{8\theta} \right)^2 + \left( e^{8\theta} \right)^2} \, d\theta \]
\[ L = \int_{0}^{2\pi} \sqrt{64e^{16\theta} + e^{16\theta}} \, d\theta \]
\[ L = \int_{0}^{2\pi} \sqrt{65e^{16\theta}} \, d\theta \]
\[ L = \int_{0}^{2\pi} \sqrt{65](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ecf1484-f95f-4119-9858-43c8382b92c0%2F07f9bccc-16cc-49ef-b318-facc7658e593%2Fansb5i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Example Problem: Finding the Length of a Polar Curve**
In this example, we are tasked with finding the exact length of a polar curve defined by the function \( r = e^{8\theta} \), where \( 0 \leq \theta \leq 2\pi \).
### Problem Statement:
**Find the exact length of the polar curve.**
\[ r = e^{8\theta}, \quad 0 \leq \theta \leq 2\pi \]
There is a box below the problem statement where the calculated numeric answer can be written.
### Solution Approach:
To find the length of a polar curve, you can use the formula for the arc length \( L \) of a curve given in polar coordinates:
\[ L = \int_{\alpha}^{\beta} \sqrt{\left( \frac{dr}{d\theta} \right)^2 + r^2} \, d\theta \]
Here:
- \( r = e^{8\theta} \)
- \( \alpha = 0 \)
- \( \beta = 2\pi \)
We need to take the following steps:
1. Calculate \( \frac{dr}{d\theta} \).
2. Substitute \( r \) and \( \frac{dr}{d\theta} \) into the arc length formula.
3. Evaluate the integral from \( \theta = 0 \) to \( \theta = 2\pi \).
### Details:
1. Calculate \( \frac{dr}{d\theta} \):
\[ \frac{dr}{d\theta} = \frac{d}{d\theta} (e^{8\theta}) = 8e^{8\theta} \]
2. Substitute into the arc length formula:
\[ L = \int_{0}^{2\pi} \sqrt{\left( 8e^{8\theta} \right)^2 + \left( e^{8\theta} \right)^2} \, d\theta \]
\[ L = \int_{0}^{2\pi} \sqrt{64e^{16\theta} + e^{16\theta}} \, d\theta \]
\[ L = \int_{0}^{2\pi} \sqrt{65e^{16\theta}} \, d\theta \]
\[ L = \int_{0}^{2\pi} \sqrt{65
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