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$L\=\left[x\+y\=0\,x\+y\=2\,2 y-3 x\=0\,2 y-3 x\=7\right]$$\stackrel{\text{assign}}{\to }$

$f\={\left(x\+y\right)}^2$$\stackrel{\text{assign}}{\to }$

$U\=x\+y$$\stackrel{\text{assign}}{\to }$

$V\=2 y-3 x$$\stackrel{\text{assign}}{\to }$$$

Obtain the vertices B, C, D of the parallelogram forming the region $R$ 

$L_1\,L_4$ = $x\+y\=0\,2y-3x\=7$$\stackrel{\text{solve}}{\to }$$\left\{x\=-\frac{7}{5}\,y\=\frac{7}{5}\right\}$$\stackrel{\text{assign to a name}}{\to }$$b$

$L_4\,L_2$ = $2y-3x\=7\,x\+y\=2$$\stackrel{\text{solve}}{\to }$$\left\{x\=-\frac{3}{5}\,y\=\frac{13}{5}\right\}$$\stackrel{\text{assign to a name}}{\to }$$c$

$L_2\,L_3$ = $x\+y\=2\,2y-3x\=0$$\stackrel{\text{solve}}{\to }$$\left\{x\=\frac{4}{5}\,y\=\frac{6}{5}\right\}$$\stackrel{\text{assign to a name}}{\to }$$d$

Implement the integration over the region $R$ 

${\int }_{-7\/5}^{-3\/5}{\int }_{-x}^{\left(3 x\+7\right)\/2}f \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_{-3\/5}^0{\int }_{-x}^{2-x}f \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_0^{4\/5}{\int }_{3 x\/2}^{2-x}f \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $\frac{56}{15}$$$

Obtain the equations $S\=\left\{x\=x\left(u\,v\right)\,y\=y\left(u\,v\right)\right\}$ for the mapping $\left(u\,v\right)\to \left(x\,y\right)$ 

$u\=U\,v\=V$

$u=x+y,v=2y-3x$

$\stackrel{\text{solve (specified)}}{\to }$

$\left\{x=\frac{2u}{5}-\frac{v}{5}\,y=\frac{v}{5}+\frac{3u}{5}\right\}$

$\stackrel{\text{assign to a name}}{\to }$

$S$

Obtain the Jacobian matrix and the Jacobian

$\genfrac{}{}{0ex}{}{x}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{S}$ = $\frac{2u}{5}-\frac{v}{5}$$\stackrel{\text{assign to a name}}{\to }$$X$$$$$

$\genfrac{}{}{0ex}{}{y}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{S}$ = $\frac{v}{5}+\frac{3u}{5}$$\stackrel{\text{assign to a name}}{\to }$$Y$$$

$\left[X\,Y\right]$ = $\left[\frac{2u}{5}-\frac{v}{5}\,\frac{v}{5}+\frac{3u}{5}\right]$$\stackrel{\text{Jacobian}}{\to }$$\frac{1}{5}$

Obtain the Jacobian matrix and the Jacobian from first principles

$\left[\begin{array}{cc}\frac{\partial }{\partial u} \mathrm{X}& \frac{\partial }{\partial v} \mathrm{X}\\ \frac{\partial }{\partial u} \mathrm{Y}& \frac{\partial }{\partial v} \mathrm{Y}\end{array}\right]$ = $\stackrel{\text{determinant}}{\to }$$\frac{1}{5}$$$

Obtain the images of the edges of the parallelogram defining region $R$ 

Transform the integrand

Expression palette: Evaluation template
Evaluate $f$ using the equations in set $S$.

Context Panel: Assign to a Name≻$F$

$\genfrac{}{}{0ex}{}{f}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{S}$ = $u^2$$\stackrel{\text{assign to a name}}{\to }$$F$$$

Implement the integration over the region $R\prime$ 

${\int }_0^2{\int }_0^{7}F\/5 \mathit{\ⅆ}v \mathit{\ⅆ}u$ = $\frac{56}{15}$$$

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

$L≔\left[x\+y\=0\,x\+y\=2\,2 y-3 x\=0\,2 y-3 x\=7\right]\:$

$f≔{\left(x\+y\right)}^2\:$

$$\begin{gathered} U≔\left(x\,y\right)\to x\+y\: \\ V≔\left(x\,y\right)\to 2 y-3 x\: \end{gathered}$$

Obtain the vertices B, C, D of the parallelogram forming the region $R$ 

$\mathrm{solve}\left(\left\{L_1\,L_4\right\}\right)$ = $\left\{x\=-\frac{7}{5}\,y\=\frac{7}{5}\right\}$$$

$\mathrm{solve}\left(\left\{L_4\,L_2\right\}\right)$ = $\left\{x\=-\frac{3}{5}\,y\=\frac{13}{5}\right\}$$$

$\mathrm{solve}\left(\left\{L_2\,L_3\right\}\right)$ = $\left\{x\=\frac{4}{5}\,y\=\frac{6}{5}\right\}$$$$$

Implement the integration over the region $R$ 

$$\begin{gathered} Q≔\mathrm{Int}\left(f\,\left[y\=-x..\left(3 x\+7\right)\/2\,x\=-7\/5..-3\/5\right]\right)\+\mathrm{Int}\left(f\,\left[y\=-x..2-x\,x\=-3\/5..0\right]\right)\+\mathrm{Int}\left(f\,\left[y\=3 x\/2..2-x\,x\=0..4\/5\right]\right)\; \\ \mathrm{value}\left(Q\right) \end{gathered}$$

${\∫}_{-\frac{7}{5}}^{-\frac{3}{5}}{\∫}_{-x}^{\frac{3}{2}x\+\frac{7}{2}}{\left(x\+y\right)}^2\ⅆy\ⅆx\+{\∫}_{-\frac{3}{5}}^0{\∫}_{-x}^{2-x}{\left(x\+y\right)}^2\ⅆy\ⅆx\+{\∫}_0^{\frac{4}{5}}{\∫}_{\frac{3}{2}x}^{2-x}{\left(x\+y\right)}^2\ⅆy\ⅆx$

Obtain the equations $S\=\left\{x\=x\left(u\,v\right)\,y\=y\left(u\,v\right)\right\}$ for the mapping $\left(u\,v\right)\to \left(x\,y\right)$ 

$S≔\mathrm{solve}\left(\left\{u\=U\left(x\,y\right)\,v\=V\left(x\,y\right)\right\}\,\left\{x\,y\right\}\right)$

$\left\{x\=\frac{2}{5}u-\frac{1}{5}v\,y\=\frac{1}{5}v\+\frac{3}{5}u\right\}$

Obtain the Jacobian matrix and the Jacobian

$$\begin{gathered} T≔\mathrm{eval}\left(\left[x\,y\right]\,S\right)\; \\ \mathrm{Jacobian}\left(T\,\left[u\,v\right]\right)\; \\ \mathrm{Jacobian}\left(T\,\left[u\,v\right]\,\mathrm{output}\=\mathrm{determinant}\right) \end{gathered}$$

$\left[\frac{2}{5}u-\frac{1}{5}v\,\frac{1}{5}v\+\frac{3}{5}u\right]$

Obtain the images of the edges of the parallelogram defining region $R$ 

Transform the integrand

$F≔\mathrm{eval}\left(f\,S\right)$

$u^2$

Implement the integration over the region $R\prime$ 

$$\begin{gathered} q≔\mathrm{Int}\left( F\/5\,\left[v\=0..7\,u\=0..2\right]\right)\; \\ \mathrm{value}\left(q\right) \end{gathered}$$

${\∫}_0^2{\∫}_0^7\frac{1}{5}{u}^2\ⅆv\ⅆu$