Define $\left(\nabla \times \mathbf{F}\right)·\mathbf{k}\=2\left(x y\+1\right)$ as $Q$

$2\left(x y\+1\right)$$\stackrel{\text{assign to a name}}{\to }$$Q$

Integrate $Q$ over the region $R$ 

${\int }_0^{1\/5}{\int }_{x^2}^{\sqrt{x}}Q \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_{1\/5}^{1\/2}{\int }_{2\/5}^{\sqrt{x}}Q \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_{1\/5}^{1\/2}{\int }_{x^2}^{3\/10}Q \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_{1\/2}^1{\int }_{x^2}^{\sqrt{x}}Q \mathit{\ⅆ}y \mathit{\ⅆ}x$

${\∫}_0^{\frac{1}{5}}{\∫}_{x^2}^{\sqrt{x}}\left(2xy\+2\right)\ⅆy\ⅆx\+{\∫}_{\frac{1}{5}}^{\frac{1}{2}}{\∫}_{\frac{2}{5}}^{\sqrt{x}}\left(2xy\+2\right)\ⅆy\ⅆx\+{\∫}_{\frac{1}{5}}^{\frac{1}{2}}{\∫}_{x^2}^{\frac{3}{10}}\left(2xy\+2\right)\ⅆy\ⅆx\+{\∫}_{\frac{1}{2}}^1{\∫}_{x^2}^{\sqrt{x}}\left(2xy\+2\right)\ⅆy\ⅆx$

$\=$

$\frac{45959}{60000}$

Table 9.10.10(a)   Integral of $Q$ over the region $R$ 

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Define the vector field F

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

Obtain the line integral of F along $y_1$ and $y_2$ 

$$\begin{gathered} \mathrm{LineInt}\left(\mathbf{F}\,\mathrm{Path}\left(⟨x\,x^2⟩\,x\=0..1\right)\,\mathrm{output}\=\mathrm{integral}\right)\+ \\ \mathrm{LineInt}\left(\mathbf{F}\,\mathrm{Path}\left(⟨x\,\sqrt{x}⟩\,x\=1..0\right)\,\mathrm{output}\=\mathrm{integral}\right) \end{gathered}$$

${\∫}_0^1\left(2x^5-2x^2\+x\right)\ⅆx\+{\∫}_1^0\left(x-2\sqrt{x}\+\frac{1}{2}{x}^2\right)\ⅆx$

$\=$

$\frac{5}{6}$

Table 9.10.10(b)   Line Integral of F along $y_1$ and $y_2$ 

$$\begin{gathered} \mathrm{LineInt}\left(\mathbf{F}\,\mathrm{Path}\left(⟨x\,3\/10⟩\,x\=1\/2..1\/5\right)\,\mathrm{output}\=\mathrm{integral}\right)\+ \\ \mathrm{LineInt}\left(\mathbf{F}\,\mathrm{Path}\left(⟨1\/5\,y⟩\,y\=3\/10..2\/5\right)\,\mathrm{output}\=\mathrm{integral}\right)\+ \\ \mathrm{LineInt}\left(\mathbf{F}\,\mathrm{Path}\left(⟨x\,2\/5⟩\,x\=1\/5..1\/2\right)\,\mathrm{output}\=\mathrm{integral}\right)\+ \\ \mathrm{LineInt}\left(\mathbf{F}\,\mathrm{Path}\left(⟨1\/2\,y⟩\,y\=2\/5..3\/10\right)\,\mathrm{output}\=\mathrm{integral}\right) \end{gathered}$$

${\∫}_{\frac{1}{2}}^{\frac{1}{5}}\left(x-\frac{3}{5}\right)\ⅆx\+{\∫}_{\frac{3}{10}}^{\frac{2}{5}}\frac{1}{25}y\ⅆy\+{\∫}_{\frac{1}{5}}^{\frac{1}{2}}\left(x-\frac{4}{5}\right)\ⅆx\+{\∫}_{\frac{2}{5}}^{\frac{3}{10}}\frac{1}{4}y\ⅆy$

$\=$

$-\frac{1347}{20000}$

Table 9.10.10(c)   Line integral of F along the bounding curves $C_k\,k\=1\,\dots \,4$