${\int }_{\frac{1-\sqrt{5}}{2}}^{\frac{1\+\sqrt{5}}{2}}{\int }_{x^2}^{x\+1}\left(1\+x^2\right) \mathrm{dy} \mathrm{dx}\=5 \sqrt{5}\/4$ 

Initialize

Loading Student:-VectorCalculus

Define the Cartesian vector field F

$⟨x-2 y\,x^{2}y⟩$ = $\stackrel{\text{to Vector Field}}{\to }$ $\stackrel{\text{assign to a name}}{\to }$$F$$$

Compute the divergence of F

$\nabla ·\mathbf{F}$ = $x^2\+1$$$

Obtain the $x$-coordinates of the intersections of $y\=x^2$ and $y\=x\+1$

$x^2\=1\+x$$\stackrel{\text{solutions for x}}{\to }$$\frac{1}{2}\sqrt{5}\+\frac{1}{2}\,\frac{1}{2}-\frac{1}{2}\sqrt{5}$$\stackrel{\text{assign to a name}}{\to }$$X$

Form, evaluate, and simplify the integral of the divergence

Loading Student:-MultivariateCalculus

$1\+x^2$

$x^2\+1$

$\stackrel{\text{MultiInt}}{\to }$

${\∫}_{\frac{1}{2}-\frac{1}{2}\sqrt{5}}^{\frac{1}{2}\sqrt{5}\+\frac{1}{2}}{\∫}_{x^2}^{1\+x}\left(x^2\+1\right)\ⅆy\ⅆx$

$\=$

$-\frac{1}{5}{\left(\frac{1}{2}\sqrt{5}\+\frac{1}{2}\right)}^5\+\frac{1}{5}{\left(\frac{1}{2}-\frac{1}{2}\sqrt{5}\right)}^5\+\frac{1}{4}{\left(\frac{1}{2}\sqrt{5}\+\frac{1}{2}\right)}^4-\frac{1}{4}{\left(\frac{1}{2}-\frac{1}{2}\sqrt{5}\right)}^4\+\frac{1}{2}{\left(\frac{1}{2}\sqrt{5}\+\frac{1}{2}\right)}^2-\frac{1}{2}{\left(\frac{1}{2}-\frac{1}{2}\sqrt{5}\right)}^2\+\sqrt{5}$

$\stackrel{\text{simplify}}{\=}$

$\frac{5}{4}\sqrt{5}$

Table 9.10.6(a)   $\nabla ·\mathbf{F}$ integrated over the region $R$ 

Compute ${∳}_{C}f \mathrm{dy}-g \mathrm{dx}$, the flux of F through $C$, the boundary of $R$ 

${\int }_{X_2}^{X_1}\genfrac{}{}{0ex}{}{\left(F_1\cdot \left(2 x\right)-F_2\right)}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{y\=x^2} \mathit{\ⅆ}x$$\stackrel{\text{simplify}}{\=}$$-\frac{8}{3}\sqrt{5}$ $$$$

${\int }_{X_1}^{X_2}\genfrac{}{}{0ex}{}{\left(F_1-F_2\right)}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{y\=1\+x}\mathit{\ⅆ}x$$\stackrel{\text{simplify}}{\=}$$\frac{47}{12}\sqrt{5}$$$

Table 9.10.6(b)   Flux of F through the boundary of $R$ 

Initialize

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{VectorCalculus}\right)\:$

$\mathrm{BasisFormat}\left(\mathrm{false}\right)\:$

$\mathbf{F}≔\mathrm{VectorField}\left(⟨x-2 y\,x^{2}y⟩\right)\:$

$X≔\mathrm{solve}\left(x^2\=x\+1\,x\right)$

$\frac{1}{2}\+\frac{1}{2}\sqrt{5}\,\frac{1}{2}-\frac{1}{2}\sqrt{5}$

Use the int command to integrate $\nabla ·\mathbf{F}$ over the region $R$ 

$\mathrm{int}\left(\mathrm{Divergence}\left(\mathbf{F}\right)\,\left[x\,y\right]\=\mathrm{Region}\left( X_2..X_1\,x^2..x\+1\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_{\frac{1}{2}-\frac{1}{2}\sqrt{5}}^{\frac{1}{2}\+\frac{1}{2}\sqrt{5}}{\∫}_{x^2}^{x\+1}\left(x^2\+1\right)\ⅆy\ⅆx$

$\mathrm{simplify}\left(\mathrm{int}\left(\mathrm{Divergence}\left(\mathbf{F}\right)\,\left[x\,y\right]\=\mathrm{Region}\left( X_2..X_1\,x^2..x\+1\right)\right)\right)$ = $\frac{5}{4}\sqrt{5}$$$

Use the Flux command to obtain the flux of F through $y\=x^2$ 

$\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Path}\left(⟨x\,x^2⟩\,x\=X_2..X_1\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_{\frac{1}{2}-\frac{1}{2}\sqrt{5}}^{\frac{1}{2}\+\frac{1}{2}\sqrt{5}}\left(2\left(-2x^2\+x\right)x-x^4\right)\ⅆx$

$Q_1≔\mathrm{simplify}\left(\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Path}\left(⟨x\,x^2⟩\,x\=X_2..X_1\right)\right)\right)$ = $-\frac{8}{3}\sqrt{5}$$$

Use the Flux command to obtain the flux of F through $y\=x\+1$  

$\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Path}\left(⟨x\,x\+1⟩\,x\=X_1..X_2\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_{\frac{1}{2}\+\frac{1}{2}\sqrt{5}}^{\frac{1}{2}-\frac{1}{2}\sqrt{5}}\left(-x-2-x^2\left(x\+1\right)\right)\ⅆx$

$Q_2≔\mathrm{simplify}\left(\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Path}\left(⟨x\,x\+1⟩\,x\=X_1..X_2\right)\right)\right)$ = $\frac{47}{12}\sqrt{5}$$$

Obtain the total flux through the boundaries of the region $R$ 

$Q_1\+Q_2$ = $\frac{5}{4}\sqrt{5}$$$