Evaluate s(t) = ||r' (u)|| du for the Bernoulli spiral r(t) = (e' cos(5t), e' sin(5t)). It is convenient to take-oo as the lower limit since s(-∞o) = 0. Then use s to obtain an arc length parametrization r₁(s) of r(t). r₁(s) = (x(s), y(s)) (Use symbolic notation and fractions where needed.) r₁(s) =
Evaluate s(t) = ||r' (u)|| du for the Bernoulli spiral r(t) = (e' cos(5t), e' sin(5t)). It is convenient to take-oo as the lower limit since s(-∞o) = 0. Then use s to obtain an arc length parametrization r₁(s) of r(t). r₁(s) = (x(s), y(s)) (Use symbolic notation and fractions where needed.) r₁(s) =
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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![**Title: Arc Length Parametrization of the Bernoulli Spiral**
Evaluate \( s(t) = \int_{-\infty}^{t} \| \mathbf{r}'(u) \| \, du \) for the Bernoulli spiral \( \mathbf{r}(t) = \langle e^t \cos(5t), \, e^t \sin(5t) \rangle \).
It is convenient to take \(-\infty\) as the lower limit since \(s(-\infty) = 0\). Then use \(s\) to obtain an arc length parametrization \( \mathbf{r}_1(s) \) of \( \mathbf{r}(t) \).
\[ \mathbf{r}_1(s) = \langle x(s), y(s) \rangle \]
(Use symbolic notation and fractions where needed.)
\[ \mathbf{r}_1(s) = \boxed{} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff95e890f-a45d-49ff-801b-f76192665ba7%2F401aa43a-e36c-4995-b823-2c1d01a939f4%2Fgj58zgd_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Arc Length Parametrization of the Bernoulli Spiral**
Evaluate \( s(t) = \int_{-\infty}^{t} \| \mathbf{r}'(u) \| \, du \) for the Bernoulli spiral \( \mathbf{r}(t) = \langle e^t \cos(5t), \, e^t \sin(5t) \rangle \).
It is convenient to take \(-\infty\) as the lower limit since \(s(-\infty) = 0\). Then use \(s\) to obtain an arc length parametrization \( \mathbf{r}_1(s) \) of \( \mathbf{r}(t) \).
\[ \mathbf{r}_1(s) = \langle x(s), y(s) \rangle \]
(Use symbolic notation and fractions where needed.)
\[ \mathbf{r}_1(s) = \boxed{} \]
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