### Evaluating Arc Length Parametrization for the Bernoulli Spiral To find the arc length parametrization of the Bernoulli spiral given by \( r(t) = \langle e^t \cos(5t), e^t \sin(5t) \rangle \), we follow these steps: First, evaluate the integral: \[ s(t) = \int_{-\infty}^{t} \| r'(u) \| \, du \] for the Bernoulli spiral. It is convenient to take \( -\infty \) as the lower limit since \( s(-\infty) = 0 \). Then, use \( s \) to obtain an arc length parametrization \( r_1(s) \) of \( r(t) \). \[ r_1(s) = \langle x(s), y(s) \rangle \] (Use symbolic notation and fractions where needed.) Fill in your answer below: \[ r_1(s) = \framebox[300pt]{ } \] ### Explanation: #### Step-by-Step Approach: 1. **Find the derivative \( r'(t) \)**. 2. **Calculate the magnitude \( \| r'(t) \| \)**. 3. **Set up the integral for \( s(t) \)** and solve it. 4. **Express \( t \) as a function of \( s \)**. 5. **Obtain \( x(s) \) and \( y(s) \)** from the parametric equations of the original spiral \( r(t) \). Upon solving this integral, you will produce the arc length parametrization \( r_1(s) \), which represents the curve based on the arc length \( s \), rather than the original parameter \( t \). This process provides a deeper understanding of plane curves and their properties in terms of arc length—a fundamental concept in calculus and differential geometry.
### Evaluating Arc Length Parametrization for the Bernoulli Spiral To find the arc length parametrization of the Bernoulli spiral given by \( r(t) = \langle e^t \cos(5t), e^t \sin(5t) \rangle \), we follow these steps: First, evaluate the integral: \[ s(t) = \int_{-\infty}^{t} \| r'(u) \| \, du \] for the Bernoulli spiral. It is convenient to take \( -\infty \) as the lower limit since \( s(-\infty) = 0 \). Then, use \( s \) to obtain an arc length parametrization \( r_1(s) \) of \( r(t) \). \[ r_1(s) = \langle x(s), y(s) \rangle \] (Use symbolic notation and fractions where needed.) Fill in your answer below: \[ r_1(s) = \framebox[300pt]{ } \] ### Explanation: #### Step-by-Step Approach: 1. **Find the derivative \( r'(t) \)**. 2. **Calculate the magnitude \( \| r'(t) \| \)**. 3. **Set up the integral for \( s(t) \)** and solve it. 4. **Express \( t \) as a function of \( s \)**. 5. **Obtain \( x(s) \) and \( y(s) \)** from the parametric equations of the original spiral \( r(t) \). Upon solving this integral, you will produce the arc length parametrization \( r_1(s) \), which represents the curve based on the arc length \( s \), rather than the original parameter \( t \). This process provides a deeper understanding of plane curves and their properties in terms of arc length—a fundamental concept in calculus and differential geometry.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Evaluating Arc Length Parametrization for the Bernoulli Spiral
To find the arc length parametrization of the Bernoulli spiral given by \( r(t) = \langle e^t \cos(5t), e^t \sin(5t) \rangle \), we follow these steps:
First, evaluate the integral:
\[ s(t) = \int_{-\infty}^{t} \| r'(u) \| \, du \]
for the Bernoulli spiral.
It is convenient to take \( -\infty \) as the lower limit since \( s(-\infty) = 0 \). Then, use \( s \) to obtain an arc length parametrization \( r_1(s) \) of \( r(t) \).
\[ r_1(s) = \langle x(s), y(s) \rangle \]
(Use symbolic notation and fractions where needed.)
Fill in your answer below:
\[ r_1(s) = \framebox[300pt]{ } \]
### Explanation:
#### Step-by-Step Approach:
1. **Find the derivative \( r'(t) \)**.
2. **Calculate the magnitude \( \| r'(t) \| \)**.
3. **Set up the integral for \( s(t) \)** and solve it.
4. **Express \( t \) as a function of \( s \)**.
5. **Obtain \( x(s) \) and \( y(s) \)** from the parametric equations of the original spiral \( r(t) \).
Upon solving this integral, you will produce the arc length parametrization \( r_1(s) \), which represents the curve based on the arc length \( s \), rather than the original parameter \( t \).
This process provides a deeper understanding of plane curves and their properties in terms of arc length—a fundamental concept in calculus and differential geometry.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F38afbefc-9f3a-4d71-a003-6e77b60fd3e9%2Fe2b0ceb5-8998-4bc4-81f2-053f0c903fee%2F1qf0qq5_processed.png&w=3840&q=75)
Transcribed Image Text:### Evaluating Arc Length Parametrization for the Bernoulli Spiral
To find the arc length parametrization of the Bernoulli spiral given by \( r(t) = \langle e^t \cos(5t), e^t \sin(5t) \rangle \), we follow these steps:
First, evaluate the integral:
\[ s(t) = \int_{-\infty}^{t} \| r'(u) \| \, du \]
for the Bernoulli spiral.
It is convenient to take \( -\infty \) as the lower limit since \( s(-\infty) = 0 \). Then, use \( s \) to obtain an arc length parametrization \( r_1(s) \) of \( r(t) \).
\[ r_1(s) = \langle x(s), y(s) \rangle \]
(Use symbolic notation and fractions where needed.)
Fill in your answer below:
\[ r_1(s) = \framebox[300pt]{ } \]
### Explanation:
#### Step-by-Step Approach:
1. **Find the derivative \( r'(t) \)**.
2. **Calculate the magnitude \( \| r'(t) \| \)**.
3. **Set up the integral for \( s(t) \)** and solve it.
4. **Express \( t \) as a function of \( s \)**.
5. **Obtain \( x(s) \) and \( y(s) \)** from the parametric equations of the original spiral \( r(t) \).
Upon solving this integral, you will produce the arc length parametrization \( r_1(s) \), which represents the curve based on the arc length \( s \), rather than the original parameter \( t \).
This process provides a deeper understanding of plane curves and their properties in terms of arc length—a fundamental concept in calculus and differential geometry.
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