exercise:E32a469470: Difference between revisions
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(Created page with "<div class="d-none"><math> \newcommand{\ex}[1]{\item } \newcommand{\sx}{\item} \newcommand{\x}{\sx} \newcommand{\sxlab}[1]{} \newcommand{\xlab}{\sxlab} \newcommand{\prov}[1] {\quad #1} \newcommand{\provx}[1] {\quad \mbox{#1}} \newcommand{\intext}[1]{\quad \mbox{#1} \quad} \newcommand{\R}{\mathrm{\bf R}} \newcommand{\Q}{\mathrm{\bf Q}} \newcommand{\Z}{\mathrm{\bf Z}} \newcommand{\C}{\mathrm{\bf C}} \newcommand{\dt}{\textbf} \newcommand{\goesto}{\rightarrow}...") |
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\newcommand{\conj}[1]{\overline{#1}} | \newcommand{\conj}[1]{\overline{#1}} | ||
\newcommand{\mathds}{\mathbb} | \newcommand{\mathds}{\mathbb} | ||
\newcommand{\dilemma}[2] { | |||
\left\{ | |||
\begin{array}{ll} | |||
#1 \\ | |||
#2 | |||
\end{array} | |||
\right. | |||
} | |||
\newcommand{\trilemma}[3] { | |||
\left\{ | |||
\begin{array}{ll} | |||
#1 \\ | |||
#2 \\ | |||
#3 | |||
\end{array} | |||
\right. | |||
} | |||
</math></div> | </math></div> | ||
Find the arc lengths of the following parametrized | Find the arc lengths of the following parametrized | ||
curves. | curves. | ||
<ul style{{=}}"list-style-type:lower-alpha"><li><math>\dilemma{x = t+1,} | <ul style{{=}}"list-style-type:lower-alpha"><li><math>\dilemma{x = t+1,} | ||
{y = t^{\frac32}, & \mbox{from | {y = t^{\frac32}, & \mbox{from $(2,1)$ to $(5,8)$.}}</math></li> | ||
<li><math>\dilemma{x = t^2,} | <li><math>\dilemma{x = t^2,} | ||
{y = \frac23 (2t+1)^\frac32, & | {y = \frac23 (2t+1)^\frac32, & | ||
\mbox{from | \mbox{from $\left(x(0),y(0)\right) = (0, \frac23)$ to | ||
to | to $\left(x(4), y(4)\right) = (16,18)$.}}</math></li> | ||
<li><math>P(t) = (t^2, t^3)</math>, \quad from <math>P(0)</math> to <math>P(2)</math>.</li> | <li><math>P(t) = (t^2, t^3)</math>, \quad from <math>P(0)</math> to <math>P(2)</math>.</li> | ||
<li><math>\dilemma{x(\theta) = a \cos^3\theta, & a > 0,} | <li><math>\dilemma{x(\theta) = a \cos^3\theta, & a > 0,} | ||
{y(\theta) = a \sin^3\theta, & | {y(\theta) = a \sin^3\theta, & | ||
\mbox{from | \mbox{from $\left(x(0), y(0)\right) = (a,0)$ to | ||
$\left(x(\frac{\pi}2), y(\frac{\pi}2)\right) = (0,a)$.}}</math></li> | |||
</ul> | </ul> | ||
Latest revision as of 21:58, 25 November 2024
[math]
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\newcommand{\C}{\mathrm{\bf C}}
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\newcommand{\ddt}{\frac{d}{dt}}
\newcommand{\dydx}{\ddxof y}
\newcommand{\nxder}[3]{\frac{d^{#1}{#2}}{d{#3}^{#1}}}
\newcommand{\deriv}[2]{\frac{d^{#1}{#2}}{dx^{#1}}}
\newcommand{\dist}{\mathrm{distance}}
\newcommand{\arccot}{\mathrm{arccot\:}}
\newcommand{\arccsc}{\mathrm{arccsc\:}}
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\newcommand{\sech}{\mathrm{sech\:}}
\newcommand{\csch}{\mathrm{csch\:}}
\newcommand{\conj}[1]{\overline{#1}}
\newcommand{\mathds}{\mathbb}
\newcommand{\dilemma}[2] {
\left\{
\begin{array}{ll}
#1 \\
#2
\end{array}
\right.
}
\newcommand{\trilemma}[3] {
\left\{
\begin{array}{ll}
#1 \\
#2 \\
#3
\end{array}
\right.
}
[/math]
Find the arc lengths of the following parametrized curves.
- [math]\dilemma{x = t+1,} {y = t^{\frac32}, & \mbox{from $(2,1)$ to $(5,8)$.}}[/math]
- [math]\dilemma{x = t^2,} {y = \frac23 (2t+1)^\frac32, & \mbox{from $\left(x(0),y(0)\right) = (0, \frac23)$ to to $\left(x(4), y(4)\right) = (16,18)$.}}[/math]
- [math]P(t) = (t^2, t^3)[/math], \quad from [math]P(0)[/math] to [math]P(2)[/math].
- [math]\dilemma{x(\theta) = a \cos^3\theta, & a \gt 0,} {y(\theta) = a \sin^3\theta, & \mbox{from $\left(x(0), y(0)\right) = (a,0)$ to $\left(x(\frac{\pi}2), y(\frac{\pi}2)\right) = (0,a)$.}}[/math]