Use one of the formulas in (5) to find the area under one arch of the cycloid x = t - sint and y = 1 - cost. Work seen below.
Use one of the formulas in (5) to find the area under one arch of the cycloid x = t - sint and y = 1 - cost. Work seen below.
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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Use one of the formulas in (5) to find the area under one arch of the cycloid x = t - sint and y = 1 - cost. Work seen below.
How were the parametric equations x = 2pi - t (3) and y = 0 (4) found? This is an explanation of the problem but the work is not 100% clear.
![The text discusses parametric equations and differentiation in the context of describing curves associated with the cycloid.
### Parametric Equations of Cycloid
The cycloid curve is defined by the parametric equations:
\[ x = t - \sin t \]
\[ y = 1 - \cos t \]
Consider the curve \( C_1 \) from \( (0, 0) \) to \( (2\pi, 0) \) defined by:
\[ x = t - \sin t \quad (1) \]
\[ y = 1 - \cos t \quad (2) \]
### Differentiation
#### Differentiating Equation (1) with respect to \( t \):
\[
\frac{d}{dt}(x) = \frac{d}{dt}(t - \sin t)
\]
\[
\frac{dx}{dt} = \frac{d}{dt}(t) - \frac{d}{dt}(\sin t)
\]
\[
\frac{dx}{dt} = 1 - \cos t \quad \left\{ \because \frac{d}{dt}(t) = 1, \frac{d}{dt}(\sin t) = \cos t \right\}
\]
\[
dx = (1 - \cos t) dt
\]
#### Differentiating Equation (2) with respect to \( t \):
\[
\frac{d}{dt}(y) = \frac{d}{dt}(1 - \cos t)
\]
\[
\frac{dy}{dt} = \frac{d}{dt}(1) - \frac{d}{dt}(\cos t)
\]
\[
\frac{dy}{dt} = 0 - (-\sin t) \quad \left\{ \because \frac{d}{dt}(k) = 0, \frac{d}{dt}(\cos t) = -\sin t \right\}
\]
\[
dy = \sin t dt
\]
### Another Segment of the Cycloid
Consider a second curve \( C_2 \) as a segment from \( (2\pi, 0) \) to \( (0, 0) \) defined by:
\[ x = 2\pi - t \quad (3) \]
\[ y = 0 \quad](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd75a0cba-20f2-443d-a1e9-f3f1ac2380ac%2Fce2f62b1-2d79-492d-af72-c10e0d67453c%2Fy3to1h_processed.png&w=3840&q=75)
Transcribed Image Text:The text discusses parametric equations and differentiation in the context of describing curves associated with the cycloid.
### Parametric Equations of Cycloid
The cycloid curve is defined by the parametric equations:
\[ x = t - \sin t \]
\[ y = 1 - \cos t \]
Consider the curve \( C_1 \) from \( (0, 0) \) to \( (2\pi, 0) \) defined by:
\[ x = t - \sin t \quad (1) \]
\[ y = 1 - \cos t \quad (2) \]
### Differentiation
#### Differentiating Equation (1) with respect to \( t \):
\[
\frac{d}{dt}(x) = \frac{d}{dt}(t - \sin t)
\]
\[
\frac{dx}{dt} = \frac{d}{dt}(t) - \frac{d}{dt}(\sin t)
\]
\[
\frac{dx}{dt} = 1 - \cos t \quad \left\{ \because \frac{d}{dt}(t) = 1, \frac{d}{dt}(\sin t) = \cos t \right\}
\]
\[
dx = (1 - \cos t) dt
\]
#### Differentiating Equation (2) with respect to \( t \):
\[
\frac{d}{dt}(y) = \frac{d}{dt}(1 - \cos t)
\]
\[
\frac{dy}{dt} = \frac{d}{dt}(1) - \frac{d}{dt}(\cos t)
\]
\[
\frac{dy}{dt} = 0 - (-\sin t) \quad \left\{ \because \frac{d}{dt}(k) = 0, \frac{d}{dt}(\cos t) = -\sin t \right\}
\]
\[
dy = \sin t dt
\]
### Another Segment of the Cycloid
Consider a second curve \( C_2 \) as a segment from \( (2\pi, 0) \) to \( (0, 0) \) defined by:
\[ x = 2\pi - t \quad (3) \]
\[ y = 0 \quad
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