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Access the PathInt command through the Context Panel

$x y$$\stackrel{\text{line integral}}{\to }$${\∫}_0^1{t}^3\sqrt{4t^2\+1}\ⅆt$$\=$$\frac{5}{24}\sqrt{5}\+\frac{1}{120}$

Form and evaluate the line integral via the PathInt command

$\mathrm{PathInt}\left(x y\,\left[x\,y\right]\=\mathrm{Path}\left(⟨x \,x^2⟩\,x\=0..1\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^1{x}^3\sqrt{4x^2\+1}\ⅆx$

$\mathrm{PathInt}\left(x y\,\left[x\,y\right]\=\mathrm{Path}\left(⟨x \,x^2⟩\,x\=0..1\right)\right)$ = $\frac{5}{24}\sqrt{5}\+\frac{1}{120}$$$

Obtain a parametric representation of the parabola

$\mathbf{R}≔⟨t\,t^2⟩\:$$$

Obtain $\mathrm{\ρ}\=\sqrt{{\stackrel{\.}{x}}^2\+{\stackrel{\.}{y}}^2}\=∥\stackrel{\.}{\mathbf{R}}∥$ 

$∥\frac{\ⅆ}{\ⅆ t} \mathbf{R}∥$ = $\sqrt{4t^2\+1}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{\ρ}$$$

Form the integrand $f\left(x\left(t\right)\,y\left(t\right)\right)\cdot \mathrm{\ρ}$ and integrate with respect to $t$ 

$\left(\genfrac{}{}{0ex}{}{x y}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\={\mathbf{R}}_1\,y\={\mathbf{R}}_2}\right)\cdot \mathrm{\ρ}$

$t^3\sqrt{4t^2\+1}$

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

${\∫}_0^1{t}^3\sqrt{4t^2\+1}\ⅆt$

$\=$

$\frac{5}{24}\sqrt{5}\+\frac{1}{120}$