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

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

$\={\int }_0^{\mathrm{\π}\/4}\left[\left(\genfrac{}{}{0ex}{}{x y}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=x\left(t\right)\,y\=y\left(t\right)}\right)\cdot \frac{d}{\mathrm{dt}}y\left(t\right)-\left(\genfrac{}{}{0ex}{}{x\+y}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=x\left(t\right)\,y\=y\left(t\right)}\right)\cdot \frac{d}{\mathrm{dt}}x\left(t\right)\right] \mathrm{dt}$

$\={\int }_0^{\mathrm{\π}\/4}\left[\left(3\+2 \cos \left(t\right)\right)\left(5\+2 \sin \left(t\right)\right)\cdot \left(2 \cos \left(t\right)\right)-\left(\left(3\+2 \cos \left(t\right)\right)\+\left(5\+2 \sin \left(t\right)\right)\right)\cdot \left(-2 \sin \left(t\right)\right)\right] \mathrm{dt}$

$\=\frac{80}{3}\+\frac{19}{3}\sqrt{2}\+3\mathrm{\π}$ ≐ 45.05

Tools≻Tasks≻Browse:

Calculus - Vector≻Integration≻Flux≻2-D≻Through a Circle

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)\,5\+2 \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

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

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

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

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

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

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

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

${\∫}_0^{\frac{1}{4}\mathrm{\π}}\left(2\left(3\+2\cos \left(t\right)\right)\left(5\+2\sin \left(t\right)\right)\cos \left(t\right)\+2\left(8\+2\cos \left(t\right)\+2\sin \left(t\right)\right)\sin \left(t\right)\right)\ⅆt$

$\=$

$\frac{80}{3}\+\frac{19}{3}\sqrt{2}\+3\mathrm{\π}$

$\stackrel{\text{at 10 digits}}{\to }$

$45.04813052$

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

$$\begin{gathered} \mathrm{Flux}\left(\mathbf{F}\,\mathrm{Arc}\left(\mathrm{Circle}\left(⟨3\,5⟩\,2\right)\,0\,\mathrm{\π}\/4\right)\, \mathrm{output}\=\mathrm{integral}\right)\= \\ \mathrm{Flux}\left(\mathbf{F}\,\mathrm{Arc}\left(\mathrm{Circle}\left(⟨3\,5⟩\,2\right)\,0\,\mathrm{\π}\/4\right)\right) \end{gathered}$$

${\∫}_0^{\frac{1}{4}\mathrm{\π}}\left(2\left(3\+2\cos \left(t\right)\right)\left(5\+2\sin \left(t\right)\right)\cos \left(t\right)\+2\left(8\+2\cos \left(t\right)\+2\sin \left(t\right)\right)\sin \left(t\right)\right)\ⅆt\=\frac{80}{3}\+\frac{19}{3}\sqrt{2}\+3\mathrm{\π}$

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

$\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Path}\left(⟨3\+2 \cos \left(t\right)\,5\+2 \sin \left(t\right)⟩\,t\=0..\mathrm{\π}\/4\right)\,\mathrm{output}\=\mathrm{plot}\,\mathrm{scaling}\=\mathrm{constrained}\,\mathrm{fieldoptions}\=\left[\mathrm{grid}\=\left[5\,5\right]\,\mathrm{scaling}\=\mathrm{constrained}\right]\right)$

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 N, a unit normal vector along the path

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

Form and evaluate the requisite line integral

$$\begin{gathered} \mathrm{Int}\left(\mathrm{eval}\left(\mathrm{DotProduct}\left(\mathbf{F}\mathbf{\,}\mathbf{N}\right)\cdot \mathrm{\ρ}\,\left[x\=R_1\,y\=R_2\right]\right)\,t\=0..\mathrm{\π}\/4\right)\= \\ \mathrm{int}\left(\mathrm{eval}\left(\mathrm{DotProduct}\left(\mathbf{F}\mathbf{\,}\mathbf{N}\right)\cdot \mathrm{\rho }\,\left[x\=R_1\,y\=R_2\right]\right)\,t\=0..\mathrm{\π}\/4\right) \end{gathered}$$

${\∫}_0^{\frac{1}{4}\mathrm{\π}}\left(2\left(3\+2\cos \left(t\right)\right)\left(5\+2\sin \left(t\right)\right)\cos \left(t\right)\+2\left(8\+2\cos \left(t\right)\+2\sin \left(t\right)\right)\sin \left(t\right)\right)\ⅆt\=\frac{80}{3}\+\frac{19}{3}\sqrt{2}\+3\mathrm{\π}$