Given r′(t)=⟨sec⁡2t,−sin⁡t⟩, find the arc length of the curve r(t) on the interval [−π3]

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liannemdh · Accepted Answer

Let&amp;nbsp;y=f(x),a≤x≤b&amp;nbsp;be the given curve. The definite integral determines the arc length LL of such a curve.=∫ab1+[∫′(x)]2&amp;nbsp;dx&amp;nbsp;Let&amp;nbsp;x=g(t),y=h(t)&amp;nbsp;where c≤x≤d be the parametric equations of the curve y=f(x).After that, the curve&#039;s arc length is determined byL=∫cd(&amp;nbsp;dx&amp;nbsp;&amp;nbsp;dt&amp;nbsp;)2+(&amp;nbsp;dy&amp;nbsp;&amp;nbsp;dt&amp;nbsp;)2&amp;nbsp;dt&amp;nbsp;Here&amp;nbsp;x(t)=sec⁡2t,y(t)=−sin⁡t&amp;nbsp;where&amp;nbsp;−π/3≤x≤π/3&amp;nbsp;be the parametric equations of the curve y=f(x).Consequently, the curve&#039;s arc length is determined byL=∫−π/3π/3sec⁡2t)2+−sin⁡t)2dt=∫−π/3π/3sec2⁡2t+sin2tdt

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

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