${\int }_C\mathbf{F}·\mathbf{dr}$

$\={\int }_a^{b}f \mathrm{dx}\+g \mathrm{dy}$

$\={\int }_0^{2 \mathrm{\π}}\left[\left(\genfrac{}{}{0ex}{}{x y}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=x\left(t\right)\,y\=y\left(t\right)}\right)\cdot \frac{d}{\mathrm{dt}}x\left(t\right)\+\left(\genfrac{}{}{0ex}{}{x\+y}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=x\left(t\right)\,y\=y\left(t\right)}\right)\cdot \frac{d}{\mathrm{dt}}y\left(t\right)\right] \mathrm{dt}$

$\={\int }_0^{2 \mathrm{\π}}\left[\left(3\+2 \cos \left(t\right)\right)\left(5\+2 \sin \left(t\right)\right)\cdot 2\+\left(\left(3\+2 \cos \left(t\right)\right)\+\left(5\+2 \sin \left(t\right)\right)\right)\cdot 2\right] \mathrm{dt}$

$\={\int }_0^{2 \mathrm{\π}}\left(22\+4\cos \left(t\right)\left(5\+2\sin \left(t\right)\right)\+4\cos \left(t\right)\+4\sin \left(t\right)\right) \mathrm{dt}$

$\=-8 \mathrm{\π}$

Initialize

Loading Student:-VectorCalculus

Define the vector field F

$⟨x y\,x\+y⟩$ = $\left[\begin{array}{c}xy\\ x+y\end{array}\right]$$\stackrel{\text{to Vector Field}}{\to }$$\left[\begin{array}{c}xy\\ x+y\end{array}\right]$$\stackrel{\text{assign to a name}}{\to }$$F$$$

Obtain the line integral via the Context Panel

$\mathbf{F}$

$\stackrel{\text{line integral}}{\to }$

${\∫}_0^{2\mathrm{\π}}\left(-2\left(3\+2\cos \left(t\right)\right)\left(5\+2\sin \left(t\right)\right)\sin \left(t\right)\+2\left(8\+2\cos \left(t\right)\+2\sin \left(t\right)\right)\cos \left(t\right)\right)\ⅆt$

$\=$

$-8\mathrm{\π}$

Define the path parametrically, as a position vector

$\mathbf{R}\=⟨3\+2 \cos \left(t\right)\,5\+2 \sin \left(t\right)⟩$$\stackrel{\text{assign}}{\to }$

Obtain T, a unit tangent vector along the path

$\mathbf{R}$ = $\stackrel{\text{tangent vector}}{\to }$ $\stackrel{\text{Euclidean-normalize}}{\to }$ $\stackrel{\text{assign to a name}}{\to }$$T$$$

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

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

Obtain the integrand $\mathbf{F}·\mathbf{T} \mathrm{ds}$ 

$\mathbf{F}·\mathbf{T} \mathrm{\ρ}$

$-2\left(3\+2\cos \left(t\right)\right)\left(5\+2\sin \left(t\right)\right)\sin \left(t\right)\+2\left(8\+2\cos \left(t\right)\+2\sin \left(t\right)\right)\cos \left(t\right)$

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

${\∫}_0^{2\mathrm{\π}}\left(-2\left(3\+2\cos \left(t\right)\right)\left(5\+2\sin \left(t\right)\right)\sin \left(t\right)\+2\left(8\+2\cos \left(t\right)\+2\sin \left(t\right)\right)\cos \left(t\right)\right)\ⅆt$

$\=$

$-8\mathrm{\π}$

Initialize

$$\begin{gathered} \mathrm{with}\left(\mathrm{Student}:-\mathrm{VectorCalculus}\right)\: \\ \mathrm{BasisFormat}\left(\mathrm{false}\right)\: \end{gathered}$$

$\mathbf{F}≔\mathrm{VectorField}\left(⟨x y\,x\+y⟩\right)\:$

Apply the LineInt command

$\mathrm{LineInt}\left(\mathbf{F}\,\mathrm{Circle}\left(⟨3\,5⟩\,2\right)\,\mathrm{output}\=\mathrm{integral}\right)\=\mathrm{LineInt}\left(\mathbf{F}\,\mathrm{Circle}\left(⟨3\,5⟩\,2\right)\right)$

${\∫}_0^{2\mathrm{\π}}\left(-2\left(3\+2\cos \left(t\right)\right)\left(5\+2\sin \left(t\right)\right)\sin \left(t\right)\+2\left(8\+2\cos \left(t\right)\+2\sin \left(t\right)\right)\cos \left(t\right)\right)\ⅆt\=-8\mathrm{\π}$

Use the LineInt command to generate a graph of F and the path of integration

$\mathrm{LineInt}\left(\mathbf{F}\,\mathrm{Circle}\left(⟨3\,5⟩\,2\right)\,\mathrm{output}\=\mathrm{plot}\,\mathrm{scaling}\=\mathrm{constrained}\,\mathrm{vectoroptions}\=\left[\mathrm{width}\=.1\right]\right)$

Define the path parametrically, as the position vector R

$\mathbf{R}≔⟨3\+2 \cos \left(t\right)\,5\+2 \sin \left(t\right)⟩$

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

$\mathrm{\ρ}≔\mathrm{Norm}\left(\mathrm{diff}\left(\mathbf{R}\,t\right)\right)$

$2$

Obtain T, a unit tangent vector along the path

$\mathbf{T}≔\mathrm{TangentVector}\left(\mathbf{R}\,t\,\mathrm{normalized}\right)$

Form and evaluate the requisite line integral

$\mathrm{Int}\left(\mathrm{DotProduct}\left(\mathbf{F}\,\mathbf{T}\right)\cdot \mathrm{\ρ}\,t\=0..2 \mathrm{\π}\right)\=\mathrm{int}\left(\mathrm{DotProduct}\left(\mathbf{F}\,\mathbf{T}\right)\cdot \mathrm{\ρ}\,t\=0..2 \mathrm{\π}\right)$

${\∫}_0^{2\mathrm{\π}}\left(-2\left(3\+2\cos \left(t\right)\right)\left(5\+2\sin \left(t\right)\right)\sin \left(t\right)\+2\left(8\+2\cos \left(t\right)\+2\sin \left(t\right)\right)\cos \left(t\right)\right)\ⅆt\=-8\mathrm{\π}$