$\sqrt{{\left(\frac{d}{\mathrm{dt}} \cos \left(t\right)\right)}^2\+{\left(\frac{d}{\mathrm{dt}}\left( \sin \left(t\right)\/2\right)\right)}^2}$

$\=\sqrt{{\sin }^2\left(t\right)\+ {\cos }^2\left(t\right)\/4}$

$\=\sqrt{4 {\sin }^2\left(t\right)\+ {\cos }^2\left(t\right)}\/2$

$\=\sqrt{4-3 {\cos }^2\left(t\right)}\/2$

Initialize

Loading Student:-VectorCalculus

Access the PathInt command through the Context Panel

$x y$$\stackrel{\text{line integral}}{\to }$${\∫}_0^{\arctan \left(2\right)}\frac{1}{4}\cos \left(t\right)\sin \left(t\right)\sqrt{4{\sin \left(t\right)}^2\+{\cos \left(t\right)}^2}\ⅆt$$\=$$-\frac{1}{36}\+\frac{17}{900}\sqrt{85}$$$

Form and evaluate the line integral via the PathInt command

$\mathrm{PathInt}\left(x y\,\left[x\,y\right]\=\mathrm{Arc}\left(\mathrm{Ellipse}\left(x^2\+4 y^2-1\right)\,0\,\mathrm{\π}\/4\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^{\arctan \left(2\right)}\frac{1}{4}\cos \left(t\right)\sin \left(t\right)\sqrt{4{\sin \left(t\right)}^2\+{\cos \left(t\right)}^2}\ⅆt$

$\mathrm{PathInt}\left(x y\,\left[x\,y\right]\=\mathrm{Arc}\left(\mathrm{Ellipse}\left(x^2\+4 y^2-1\right)\,0\, \mathrm{\π}\/4\right)\right)$ = $-\frac{1}{36}\+\frac{17}{900}\sqrt{85}$$$

Define $f$ as a scalar-valued function of a vector argument

$f\left(\mathbf{v}\right)\=v_1\cdot v_2$$\stackrel{\text{assign as function}}{\to }$$f$

Define the ellipse parametrically as the position vector R

$\mathbf{R}\=⟨ \cos \left(t\right)\, \sin \left(t\right)\/2⟩$$\stackrel{\text{assign}}{\to }$

Obtain $\mathrm{\ρ}\=\frac{\mathrm{ds}}{\mathrm{dt}}\=∥\stackrel{\.}{\mathbf{R}}∥$ 

$∥\frac{\ⅆ}{\ⅆ t} \mathbf{R}∥$ = $\frac{1}{2}\sqrt{-3{\cos \left(t\right)}^2\+4}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{\ρ}$$$

Form and evaluate the line integral ${\int }_0^{\arctan \left(\sqrt{2}\right)}f\left(\mathbf{R}\right) \mathrm{\ρ} \mathrm{dt}$ 

${\int }_0^{\arctan \left(2\right)}\mathrm{\ρ} f\left(\mathbf{R}\right) \mathit{\ⅆ}t$ = $-\frac{1}{36}\+\frac{17}{900}\sqrt{85}$$$

Explicit formulation and evaluation of the line integral

$\mathrm{\ρ} f\left(\mathbf{R}\right)$

$\frac{1}{4}\sqrt{-3{\cos \left(t\right)}^2\+4}\cos \left(t\right)\sin \left(t\right)$

$\stackrel{\text{integrate w.r.t. t}}{\to }$

${\∫}_0^{\arctan \left(2\right)}\frac{1}{4}\sqrt{-3{\cos \left(t\right)}^2\+4}\cos \left(t\right)\sin \left(t\right)\ⅆt$

$\=$

$-\frac{1}{36}\+\frac{17}{900}\sqrt{85}$