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>
$$\displaystyle{x}={\cos{\cdot}}{e}^{{t}}$$
$$\displaystyle{y}={\sin{\cdot}}{e}^{{t}}$$

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
$$\displaystyle{L}={\int_{{0}}^{\pi}}{s}{q}{t}{\left({\left({\sin{{e}}}^{{t}}\right)}^{{2}}+{\left({\cos{{e}}}^{{t}}\right)}^{{2}}{\left.{d}{t}\right.}\right)}$$
$$\displaystyle={\int_{{0}}^{\pi}}{s}{q}{t}{\left({{\sin}^{{2}}{e}^{{t}}}+{{\cos}^{{2}}{e}^{{t}}}{\left.{d}{t}\right.}\right)}$$
$$\displaystyle={\int_{{0}}^{\pi}}{\left.{d}{t}\right.}$$
$$\displaystyle=\pi$$