Find the arc length of the curve r(t) on the interval [−π/3].

geduiwelh 2021-03-09 Answered

Given \(\displaystyle{r}′{\left({t}\right)}=⟨{\sec{{2}}}{t},−{\sin{{t}}}⟩\), find the arc length of the curve r(t) on the interval \(\displaystyle{\left[-\frac{\pi}{{3}}\right]}\)

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

liannemdh
Answered 2021-03-10 Author has 20825 answers

Let \(\displaystyle{y}={f{{\left({x}\right)}}},{a}\leq{x}\leq{b}\) be the given curve. The arc length LL of such curve is given by the definite integral
\(\displaystyle={\int_{{{a}}}^{{{b}}}}\sqrt{{{1}+{\left[\int'{\left({x}\right)}\right]}^{{{2}}}}}{\left.{d}{x}\right.}\)
Let \(\displaystyle{x}={g{{\left({t}\right)}}},{y}={h}{\left({t}\right)}\) where c≤x≤d be the parametric equations of the curve y=f(x).
Then the arc length of the curve is given by
\(\displaystyle{L}={\int_{{{c}}}^{{{d}}}}\sqrt{{{\left({\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}\right)}^{{2}}+{\left({\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{t}\right.}}}}\right)}^{{2}}}}{\left.{d}{t}\right.}\)
Here \(x(t)=\sec2t,y(t)=−\sin t\) where \(−π/3≤x≤π/3\) be the parametric equations of the curve y=f(x).

Then the arc length of the curve is given by

\(L=\int_{-π/3}^{π/3}\sqrt{\sec2t)^2+-\sin t)^2}dt =\int_{-π/3}^{π/3}\sqrt{\sec^22t+\sin^2}tdt\)

Not exactly what you’re looking for?
Ask My Question
46
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-06-05

Find the exact length of the curve. Use a graph to determine the parameter interval.
\(r=\cos^2(\frac{\theta}{2})\)

asked 2021-01-08

Find, correct to four decimal places, the length of the curve of intersection of the cylinder \(\displaystyle{4}{x}^{{2}}+{y}^{{2}}={4}\) and the plane \(x + y + z = 5.\)

asked 2021-02-24

Graph the plane curve defined by the parametric equations \(x = 125 \cos t\ and\ y = 125 \sin t\)

asked 2021-02-02

Let \(\displaystyle{f{{\left({x},{y}\right)}}}=-\frac{{{x}{y}}}{{{x}^{{2}}+{y}^{{2}}}}\).
Find limit of \(f(x,y)\ \text{as}\ (x,y)\ \rightarrow (0,0)\ \text{i)Along y axis and ii)along the line}\ y=x.\ \text{Evaluate Limes}\ \lim_{x,y\rightarrow(0,0)}y\log(x^{2}+y^{2})\),by converting to polar coordinates.

asked 2020-11-17

Write a short paragraph explaining this statement. Use the following example and your answers How long does it take the particle to go once around the circle? Find parametric equations if the particle moves twice as fast around the circle. The position of a particle is given by the parametric equations \(x = \sin t, y = \cos t\) where 1 represents time. We know that the shape of the path of the particle is a circle.

asked 2020-12-16

Find the gradient at the point (3,-3,2) of the scaler field given by \(f=xy-4ze+35\)

asked 2021-02-25

Let \(\displaystyle{A}={\left\lbrace{x}:{x}{A}={\left\lbrace{x}:{x}\ {i}{s}\ {a}\ {n}{a}{t}{u}{r}{a}{l}\ nu{m}{b}{e}{r}{\quad\text{and}\quad}{a}\ {f}{a}{c}to{r}{\ o}{f}\ {18}\right\rbrace}\right.}\)
\(\displaystyle{B}={\left\lbrace{x}:{x}{B}={\left\lbrace{x}:{x}\ {i}{s}\ {a}\ {n}{a}{t}{u}{r}{a}{l}\ nu{m}{b}{e}{r}{\quad\text{and}\quad}\ le{s}{s}\ {t}{h}{a}{n}\ {6}\right\rbrace}\right.}\)
Find \(\displaystyle{A}\cup{B}{\quad\text{and}\quad}{A}\cap{B}.\)

...