${\int }_{{\sqrt{\frac{3}{26}}}_{}}^{\frac{1}{\sqrt{2}}}{\int }_{x_2}^{x_3}f \mathrm{dx} \mathrm{dy}\+{\int }_{\frac{1}{5}}^{{\sqrt{\frac{3}{26}}}_{}}{\int }_{x_2}^{x_1}f \mathrm{dx} \mathrm{dy}$ = 2

$\int {\int }_{R\prime }f\left(x\left(u\,v\right)\,y\left(u\,v\right)\right)\left|\frac{\partial \left(x\,y\right)}{\partial \left(u\,v\right)}\right| \mathrm{dv} \mathrm{du}$

= $\frac{1}{2}{\int }_1^2{\int }_{2 u-1}^{6 u-5}\mathrm{dv} \mathrm{du}\+\frac{1}{2}{\int }_2^3{\int }_{2 u-1}^{11-2 u}\mathrm{dv} \mathrm{du}$

$\=2$

Initialize

Loading Student:-MultivariateCalculus

$L\=\left[16x^{2}y\+16y^3\+8x-88 y\=0\,144x^{2}y\+144y^3-24 x-120 y\=0\,16 x^{2}y\+16y^3-8 x-8 y\=0\right]$$\stackrel{\text{assign}}{\to }$

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

$U\=x^2\+y^2$$\stackrel{\text{assign}}{\to }$

$V\=x\/y$$\stackrel{\text{assign}}{\to }$

$a$

$L_1\,L_3$

$16x^{2}y\+16y^3\+8x-88y\=0\,16x^{2}y\+16y^3-8x-8y\=0$

$\stackrel{\text{solve}}{\to }$

$\left\{x\=0\,y\=0\right\}\,\left\{x\=\frac{5}{26}\sqrt{78}\,y\=\frac{1}{26}\sqrt{78}\right\}\,\left\{x\=-\frac{5}{26}\sqrt{78}\,y\=-\frac{1}{26}\sqrt{78}\right\}$

$b$

$L_1\,L_2$

$16x^{2}y\+16y^3\+8x-88y\=0\,144x^{2}y\+144y^3-24x-120y\=0$

$\stackrel{\text{solve}}{\to }$

$\left\{x\=0\,y\=0\right\}\,\left\{x\=\frac{7}{5}\,y\=\frac{1}{5}\right\}\,\left\{x\=-\frac{7}{5}\,y\=-\frac{1}{5}\right\}$

$c$

$L_1\,L_3$

$16x^{2}y\+16y^3\+8x-88y\=0\,16x^{2}y\+16y^3-8x-8y\=0$

$\stackrel{\text{solve}}{\to }$

$\left\{x\=0\,y\=0\right\}\,\left\{x\=\frac{5}{26}\sqrt{78}\,y\=\frac{1}{26}\sqrt{78}\right\}\,\left\{x\=-\frac{5}{26}\sqrt{78}\,y\=-\frac{1}{26}\sqrt{78}\right\}$

$L_1$

$16x^{2}y\+16y^3\+8x-88y\=0$

$\stackrel{\text{solutions for x}}{\to }$

$\frac{1}{4}\frac{-1\+\sqrt{-16y^4\+88y^2\+1}}{y}\,-\frac{1}{4}\frac{1\+\sqrt{-16y^4\+88y^2\+1}}{y}$

$\stackrel{\text{select entry 1}}{\to }$

$\frac{1}{4}\frac{-1\+\sqrt{-16y^4\+88y^2\+1}}{y}$

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

$e_1$

$L_2$

$144x^{2}y\+144y^3-24x-120y\=0$

$\stackrel{\text{solutions for x}}{\to }$

$\frac{1}{12}\frac{1\+\sqrt{-144y^4\+120y^2\+1}}{y}\,-\frac{1}{12}\frac{-1\+\sqrt{-144y^4\+120y^2\+1}}{y}$

$\stackrel{\text{select entry 1}}{\to }$

$\frac{1}{12}\frac{1\+\sqrt{-144y^4\+120y^2\+1}}{y}$

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

$e_2$

$L_3$

$16x^{2}y\+16y^3-8x-8y\=0$

$\stackrel{\text{solutions for x}}{\to }$

$\frac{1}{4}\frac{1\+\sqrt{-16y^4\+8y^2\+1}}{y}\,-\frac{1}{4}\frac{-1\+\sqrt{-16y^4\+8y^2\+1}}{y}$

$\stackrel{\text{select entry 1}}{\to }$

$\frac{1}{4}\frac{1\+\sqrt{-16y^4\+8y^2\+1}}{y}$

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

$e_3$

${\int }_{{\sqrt{3\/26}}_{}}^{1\/\sqrt{2}}{\int }_{e_2}^{e_3}f\left(x\,y\right) \mathit{\ⅆ}x \mathit{\ⅆ}y\+{\int }_{1\/5}^{{\sqrt{3\/26}}_{}}{\int }_{e_2}^{e_1}f\left(x\,y\right) \mathit{\ⅆ}x \mathit{\ⅆ}y$ = $2$

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

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

$u=x^2+y^2,v=\frac{x}{y}$

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

$\left\{x=v\mathrm{RootOf}\left(\left(v^2+1\right){\mathrm{\_Z}}^2-u\right)\,y=\mathrm{RootOf}\left(\left(v^2+1\right){\mathrm{\_Z}}^2-u\right)\right\}$

$\stackrel{\text{all values}}{\to }$

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

$\stackrel{\text{select entry 1}}{\to }$

$\left\{x=v\sqrt{\frac{u}{v^2+1}}\,y=\sqrt{\frac{u}{v^2+1}}\right\}$

$\stackrel{\text{simplify symbolic}}{\to }$

$\left\{x=\frac{v\sqrt{u}}{\sqrt{v^2+1}}\,y=\frac{\sqrt{u}}{\sqrt{v^2+1}}\right\}$

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

$S$

Obtain the Jacobian matrix and the Jacobian

$\genfrac{}{}{0ex}{}{x}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{S}$ = $\frac{v\sqrt{u}}{\sqrt{v^2+1}}$$\stackrel{\text{assign to a name}}{\to }$$X$

$\genfrac{}{}{0ex}{}{y}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{S}$ = $\frac{\sqrt{u}}{\sqrt{v^2+1}}$$\stackrel{\text{assign to a name}}{\to }$$Y$

$\left[X\,Y\right]$ = $\left[\frac{v\sqrt{u}}{\sqrt{v^2+1}}\,\frac{\sqrt{u}}{\sqrt{v^2+1}}\right]$$\stackrel{\text{Jacobian}}{\to }$$-\frac{1}{2\left(v^2+1\right)}$$\stackrel{\text{assign to a name}}{\to }$$J$

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}{2\left(v^2\+1\right)}$

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

Obtain the transformed integrand $f\left(x\left(u\,v\right)\,y\left(u\,v\right)\right)\left|\frac{\partial \left(x\,y\right)}{\partial \left(u\,v\right)}\right|$ 

$f\left(X\,Y\right) \left|J\right|$ = $\frac{1}{2}\frac{\left(\frac{u}{v^2\+1}\+\frac{v^{2}u}{v^2\+1}\right)\left(v^2\+1\right)}{u\left|v^2\+1\right|}$$\stackrel{\text{assuming real}}{\to }$$\frac{1}{2}$

Implement the integration over the region $R\prime$ 

${\int }_1^2{\int }_{2 u-1}^{6 u-5}\frac{1}{2} \mathit{\ⅆ}v \mathit{\ⅆ}u\+{\int }_2^3{\int }_{2 u-1}^{11-2 u}\frac{1}{2} \mathit{\ⅆ}v \mathit{\ⅆ}u$ = $2$

Initialize

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

$f≔\left(x\,y\right)\to \left(x^2\+y^2\right)\/y^2\:$

$L≔\left[16x^{2}y\+16y^3\+8x-88 y\=0\,144x^{2}y\+144y^3-24 x-120 y\=0\,16 x^{2}y\+16y^3-8 x-8 y\=0\right]\:$

$$\begin{gathered} U≔x^2\+y^2\: \\ V≔x\/y\: \end{gathered}$$

$X_1≔\mathrm{solve}\left(L_1\,x\right)\left[1\right]\:$

$X_1$ = $\frac{1}{4}\frac{-1\+\sqrt{-16y^4\+88y^2\+1}}{y}$

$X_2≔\mathrm{solve}\left(L_2\,x\right)\left[1\right]\:$

$X_2$ = $\frac{1}{12}\frac{1\+\sqrt{-144y^4\+120y^2\+1}}{y}$

$X_3≔\mathrm{solve}\left(L_3\,x\right)\left[1\right]\:$

$X_3$ = $\frac{1}{4}\frac{1\+\sqrt{-16y^4\+8y^2\+1}}{y}$

Table 5.6.5(e)   Each $C_k$ in $R$ solved for $x\=x\left(y\right)$

Point

Coordinates

$a$

$\mathrm{solve}\left(\left\{L_2\,L_3\right\}\,\mathrm{explicit}\right)$

$\left\{x\=0\,y\=0\right\}\,\left\{x\=\frac{1}{2}\sqrt{2}\,y\=\frac{1}{2}\sqrt{2}\right\}\,\left\{x\=-\frac{1}{2}\sqrt{2}\,y\=-\frac{1}{2}\sqrt{2}\right\}$

$b$

$\mathrm{solve}\left(\left\{L_1\,L_2\right\}\right)$

$\left\{x\=0\,y\=0\right\}\,\left\{x\=\frac{7}{5}\,y\=\frac{1}{5}\right\}\,\left\{x\=-\frac{7}{5}\,y\=-\frac{1}{5}\right\}$

$c$

$\mathrm{solve}\left(\left\{L_1\,L_3\right\}\,\mathrm{explicit}\right)$

$\left\{x\=0\,y\=0\right\}\,\left\{x\=\frac{5}{26}\sqrt{78}\,y\=\frac{1}{26}\sqrt{78}\right\}\,\left\{x\=-\frac{5}{26}\sqrt{78}\,y\=-\frac{1}{26}\sqrt{78}\right\}$

Table 5.6.5(f)   Corners of region $R$: Select appropriate solution based on Figure 5.6.5(a)

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

$$\begin{gathered} \mathrm{temp}≔\mathrm{solve}\left(\left\{u\=U\,v\=V\right\}\,\left\{x\,y\right\}\,\mathrm{explicit}\right)\; \\ S≔\mathrm{simplify}\left(\mathrm{temp}\left[1\right]\right) assuming \mathrm{positive} \end{gathered}$$

$\left\{x\=\frac{v\sqrt{\left(v^2\+1\right)u}}{v^2\+1}\,y\=\frac{\sqrt{\left(v^2\+1\right)u}}{v^2\+1}\right\}\,\left\{x\=-\frac{v\sqrt{\left(v^2\+1\right)u}}{v^2\+1}\,y\=-\frac{\sqrt{\left(v^2\+1\right)u}}{v^2\+1}\right\}$

Obtain the Jacobian matrix and the Jacobian

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

$\left[\frac{v\sqrt{u}}{\sqrt{v^2\+1}}\,\frac{\sqrt{u}}{\sqrt{v^2\+1}}\right]$

Obtain the images of the boundaries of the region $R$ 

Obtain the transformed integrand $f\left(x\left(u\,v\right)\,y\left(u\,v\right)\right)$ 

$\mathrm{simplify}\left(f\left(T_1\,T_2\right)\right)$ = $v^2\+1$

Implement the integration over the region $R\prime$ 

$$\begin{gathered} q≔\mathrm{Int}\left(1\/2\,\left[v\=2 u-1..6 u-5\,u\=1..2\right]\right)\+\mathrm{Int}\left(1\/2\,\left[v\=2 u-1..11-2 u\,u\=2..3\right]\right)\; \\ \mathrm{value}\left(q\right) \end{gathered}$$

${\∫}_1^2{\∫}_{2u-1}^{6u-5}\frac{1}{2}\ⅆv\ⅆu\+{\∫}_2^3{\∫}_{2u-1}^{11-2u}\frac{1}{2}\ⅆv\ⅆu$