${\int }_C\mathbf{F}\mathbf{·}\mathbf{N}\mathbf{ }\mathrm{ds}$

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

$\={\int }_0^1\left[\left(\genfrac{}{}{0ex}{}{x y}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=x\left(t\right)\,y\=y\left(t\right)}\right)\cdot \frac{d}{\mathrm{dt}}y\left(t\right)-\left(\genfrac{}{}{0ex}{}{x\+y}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=x\left(t\right)\,y\=y\left(t\right)}\right)\cdot \frac{d}{\mathrm{dt}}x\left(t\right)\right] \mathrm{dt}$

$\={\int }_0^1\left[\left(3\+2 \cos \left(t\right)\right)\left(4\+\sin \left(t\right)\right)\cdot \cos \left(t\right)-\left(\left(3\+2 \cos \left(t\right)\right)\+\left(4\+\sin \left(t\right)\right)\right)\cdot \left(-2 \sin \left(t\right)\right)\right] \mathrm{dt}$

$\=10 \mathrm{\π}$

Tools≻Tasks≻Browse:

Calculus - Vector≻Integration≻Flux≻2-D≻Through an Ellipse

Initialize

Loading Student:-VectorCalculus

Define the vector field F

$⟨x y\,x\+y⟩$ = $\stackrel{\text{to Vector Field}}{\to }$ $\stackrel{\text{assign to a name}}{\to }$$F$$$

Define the path parametrically, as a position vector

$\mathbf{R}\=⟨3\+2 \cos \left(t\right)\,4\+ \sin \left(t\right)⟩$$\stackrel{\text{assign}}{\to }$$$

Obtain $\stackrel{\mathit{\.}}{\mathbf{R}}$, a tangent vector along the path

$\frac{\ⅆ}{\ⅆ t} \mathbf{R}$ = $$

Obtain N, a unit normal vector along the path

$⟨\cos \left(t\right)\,2 \sin \left(t\right)⟩$ = $\stackrel{\text{Euclidean-normalize}}{\to }$ $\stackrel{\text{assign to a name}}{\to }$$N$$$

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

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

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

$\genfrac{}{}{0ex}{}{\mathbf{F}·\mathbf{N} \mathrm{\ρ}}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=R_1\,y\=R_2}$

$\left(\frac{\left(2\cos \left(t\right)\+3\right)\left(4\+\sin \left(t\right)\right)\cos \left(t\right)}{\sqrt{-3{\cos \left(t\right)}^2\+4}}\+\frac{2\left(2\cos \left(t\right)\+7\+\sin \left(t\right)\right)\sin \left(t\right)}{\sqrt{-3{\cos \left(t\right)}^2\+4}}\right)\sqrt{-3{\cos \left(t\right)}^2\+4}$

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

${\∫}_0^{2\mathrm{\π}}\left(\frac{\left(2\cos \left(t\right)\+3\right)\left(4\+\sin \left(t\right)\right)\cos \left(t\right)}{\sqrt{-3{\cos \left(t\right)}^2\+4}}\+\frac{2\left(2\cos \left(t\right)\+7\+\sin \left(t\right)\right)\sin \left(t\right)}{\sqrt{-3{\cos \left(t\right)}^2\+4}}\right)\sqrt{-3{\cos \left(t\right)}^2\+4}\ⅆt$

$\=$

$10\mathrm{\π}$

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 Flux command

$$\begin{gathered} \mathrm{Flux}\left(\mathbf{F}\,\mathrm{Ellipse}\left(\frac{{\left(x-3\right)}^2}{4}\+{\left(y-4\right)}^2-1\right)\,\mathrm{output}\=\mathrm{integral}\right)\= \\ \mathrm{Flux}\left(\mathbf{F}\,\mathrm{Ellipse}\left(\frac{{\left(x-3\right)}^2}{4}\+{\left(y-4\right)}^2-1\right)\right) \end{gathered}$$

${\∫}_0^{2\mathrm{\π}}\left(\left(2\cos \left(t\right)\+3\right)\left(4\+\sin \left(t\right)\right)\cos \left(t\right)\+2\left(2\cos \left(t\right)\+7\+\sin \left(t\right)\right)\sin \left(t\right)\right)\ⅆt\=10\mathrm{\π}$

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

$\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Path}\left(\mathbf{R}\,t\=0..2 \mathrm{\π}\right)\,\mathrm{output}\=\mathrm{plot}\,\mathrm{scaling}\=\mathrm{constrained}\right)$

Define the path parametrically, as the position vector R

$\mathbf{R}≔⟨3\+2 \cos \left(t\right)\,4\+ \sin \left(t\right)⟩$

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

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

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

Obtain N, a unit normal vector along the path

$\mathbf{N}≔\mathrm{Normalize}\left(⟨\cos \left(t\right)\,2 \sin \left(t\right)⟩\right)$

Form and evaluate the requisite line integral

$$\begin{gathered} \mathrm{Int}\left(\mathrm{eval}\left(\mathrm{DotProduct}\left(\mathbf{F}\mathbf{\,}\mathbf{N}\right)\cdot \mathrm{\ρ}\,\left[x\=R_1\,y\=R_2\right]\right)\,t\=0..2 \mathrm{\π}\right)\= \\ \mathrm{int}\left(\mathrm{eval}\left(\mathrm{DotProduct}\left(\mathbf{F}\mathbf{\,}\mathbf{N}\right)\cdot \mathrm{\rho }\,\left[x\=R_1\,y\=R_2\right]\right)\,t\=0..2 \mathrm{\π}\right) \end{gathered}$$

${\∫}_0^{2\mathrm{\π}}\left(\frac{\left(2\cos \left(t\right)\+3\right)\left(4\+\sin \left(t\right)\right)\cos \left(t\right)}{\sqrt{-3{\cos \left(t\right)}^2\+4}}\+\frac{2\left(2\cos \left(t\right)\+7\+\sin \left(t\right)\right)\sin \left(t\right)}{\sqrt{-3{\cos \left(t\right)}^2\+4}}\right)\sqrt{-3{\cos \left(t\right)}^2\+4}\ⅆt\=10\mathrm{\π}$