Find the exact length of the parametric curve 0&lt;t&lt;π x=cos⋅et y=sin⋅et

Let y=f(x), a≤x≤by=f(x),a≤x≤b be the given curve.The arc length L of such curve is given by the definite integralL=∫ab1+[f′(x))2dxLet x=g(t), y=h(t) where c≤x≤dc≤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)=cos⁡et,y(t)=sin⁡et where 0&lt;t&lt;π be the parametric equations of the curve y=f(x). Then the arc length of the curve is given byL=∫0πsqt((sin⁡et)2+(cos⁡et)2dt)=∫0πsqt(sin2et+cos2etdt)=∫0πdt=π

Expert Answer to "Find the exact length of the parametric curve..."

High School
Other

Kye

Answered question

2020-10-18

Find the exact length of the parametric curve $0 < t < π$
$x = \cos \cdot e^{t}$
$y = \sin \cdot e^{t}$

Answer & Explanation

Laith Petty

Skilled2020-10-19Added 103 answers

Let y=f(x), $a \leq x \leq b y = f (x), a \leq x \leq b$ be the given curve.The arc length L of such curve is given by the definite integral
$L = \int_{a}^{b} \sqrt{1 + {[f^{'} (x))}^{2}} d x$
Let x=g(t), y=h(t) where $c \leq x \leq d c \leq x \leq d$ 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 $x (t) = {\cos e}^{t}, y (t) = {\sin e}^{t}$ where $0 < t < π$ be the parametric equations of the curve y=f(x). Then the arc length of the curve is given by
$L = \int_{0}^{π} s q t ({({\sin e}^{t})}^{2} + {({\cos e}^{t})}^{2} d t)$
$= \int_{0}^{π} s q t (\sin^{2} e^{t} + \cos^{2} e^{t} d t)$
$= \int_{0}^{π} d t$
$= π$

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Other

Didn't find what you were looking for?