Given Undefined control sequence sintUndefined control sequence sint, find the arc length of the curve r(t) on the interval [−π/3].

Let y=f(x),a≤x≤b be the given curve. The arc length LL of such curve is given by the definite integral=∫ab1+[∫′(x)]2dxLet x=g(t),y=h(t) where c≤x≤d be the parametric equations of the curve y=f(x).Then the arc length of the curve is given byL=∫cd(dxdt)2+(dydt)2dtHere x(t)=sec⁡2t,y(t)=−sin⁡t where −π/3≤x≤π/3be the parametric equations of the curve y=f(x). Then the arc length of the curve is given byL=∫−π/3π/3sec⁡(2t)2+−sin⁡(t)2dt=∫−π/3π/3sec2⁡2t+sin2tdt.

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College
Calculus and Analysis
Integral Calculus
Parametric equations, polar coordinates, and vector-valued functions

avissidep

Answered question

2020-12-30

Given

Undefined control sequence \sint

, find the arc length of the curve r(t) on the interval

[- π / 3] .

Answer & Explanation

Roosevelt Houghton

Skilled2020-12-31Added 106 answers

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

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