Question

Find the exact length of the parametric curve 0<t<pi x = cos*e^t y= sin *e^t

Find the exact length of the parametric curve \(\displaystyle{0}{<}{t}{<}\pi\)</span>
\(\displaystyle{x}={\cos{\cdot}}{e}^{{t}}\)
\(\displaystyle{y}={\sin{\cdot}}{e}^{{t}}\)

Answers (1)

2020-10-19
Let y=f(x), \(\displaystyle{a}\le{x}\le{b}{y}={f{{\left({x}\right)}}},{a}\le{x}\le{b}\) be the given curve.The arc length L of such curve is given by the definite integral
\(\displaystyle{L}={\int_{{a}}^{{b}}}\sqrt{{{1}+{\left[{f}'{\left({x}\right)}\right)}^{{2}}}}{\left.{d}{x}\right.}\)
Let x=g(t), y=h(t) where \(\displaystyle{c}\le{x}\le{d}{c}\le{x}\le{d}\) 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 \(\displaystyle{x}{\left({t}\right)}={{\cos{{e}}}^{{t}},}{y}{\left({t}\right)}={{\sin{{e}}}^{{t}}}\) where \(\displaystyle{0}{<}{t}{<}\pi\)</span> be the parametric equations of the curve y=f(x). Then the arc length of the curve is given by
\(\displaystyle{L}={\int_{{0}}^{\pi}}{s}{q}{t}{\left({\left({\sin{{e}}}^{{t}}\right)}^{{2}}+{\left({\cos{{e}}}^{{t}}\right)}^{{2}}{\left.{d}{t}\right.}\right)}\)
\(\displaystyle={\int_{{0}}^{\pi}}{s}{q}{t}{\left({{\sin}^{{2}}{e}^{{t}}}+{{\cos}^{{2}}{e}^{{t}}}{\left.{d}{t}\right.}\right)}\)
\(\displaystyle={\int_{{0}}^{\pi}}{\left.{d}{t}\right.}\)
\(\displaystyle=\pi\)
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