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

$\left[\begin{array}{c}\frac{\partial }{\partial u} \end{array}\right]$