${\int }_{10}^{20}{\int }_4^{15}{\int }_1^5\frac{1}{2\sqrt{u v w}} \mathrm{du} \mathrm{dv} \mathrm{dw}$

= $20\left(\sqrt{6}-\sqrt{30}\right)\+40\left(\sqrt{2}\+\sqrt{15}-\sqrt{3}\right)\+8\left(2\sqrt{5}-\sqrt{10}\right)-80$

$\doteq 12.13$

Initialize

Loading Student:-MultivariateCalculus

Obtain the Jacobian $\frac{\partial \left(x\,y\,z\right)}{\partial \left(u\,v\,w\right)}$ 

$x y\,x z\,y z$$\stackrel{\text{Jacobian}}{\to }$$\left[\begin{array}{ccc}y& x& 0\\ z& 0& x\\ 0& z& y\end{array}\right]$$\stackrel{\text{determinant}}{\to }$$-2yxz$

Invert the transformation $u\=x y\,v\=x z\,w\=y z$ 

$u\=x y\,v\=x z\,w\=y z$

$u\=xy\,v\=xz\,w\=yz$

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

$\left\{x\=\frac{v}{\mathrm{RootOf}\left({\mathrm{\_Z}}^{2}u-vw\right)}\,y\=\frac{w}{\mathrm{RootOf}\left({\mathrm{\_Z}}^{2}u-vw\right)}\,z\=\mathrm{RootOf}\left({\mathrm{\_Z}}^{2}u-vw\right)\right\}$

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

$\left\{x\=\frac{v}{\sqrt{\frac{vw}{u}}}\,y\=\frac{w}{\sqrt{\frac{vw}{u}}}\,z\=\sqrt{\frac{vw}{u}}\right\}\,\left\{x\=-\frac{v}{\sqrt{\frac{vw}{u}}}\,y\=-\frac{w}{\sqrt{\frac{vw}{u}}}\,z\=-\sqrt{\frac{vw}{u}}\right\}$

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

$\left\{x\=\frac{v}{\sqrt{\frac{vw}{u}}}\,y\=\frac{w}{\sqrt{\frac{vw}{u}}}\,z\=\sqrt{\frac{vw}{u}}\right\}$

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

$S$

Obtain the absolute value of the Jacobian $\frac{\partial \left(u\,v\,w\right)}{\partial \left(x\,y\,z\right)}$ 

$\genfrac{}{}{0ex}{}{\frac{1}{2 x y z}}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{S}$ = $\frac{1}{2}\frac{\sqrt{\frac{vw}{u}}}{vw}$$\stackrel{\text{assuming positive}}{\to }$$\frac{1}{2\sqrt{v}\sqrt{u}\sqrt{w}}$

Use an appropriate iterated triple-integral to calculate the volume of $R$ 

${\int }_{10}^{20}{\int }_4^{15}{\int }_1^5\frac{1}{2\sqrt{u v w}} \mathit{\ⅆ}u \mathit{\ⅆ}v \mathit{\ⅆ}w$

$-20\sqrt{3}\sqrt{5}\sqrt{2}\+40\sqrt{2}\+20\sqrt{3}\sqrt{2}-8\sqrt{5}\sqrt{2}\+40\sqrt{5}\sqrt{3}-80-40\sqrt{3}\+16\sqrt{5}$

$\stackrel{\text{combine}}{\=}$

$-20\sqrt{30}\+40\sqrt{2}\+20\sqrt{6}-8\sqrt{10}\+40\sqrt{15}-80-40\sqrt{3}\+16\sqrt{5}$

$\stackrel{\text{at 10 digits}}{\to }$

$12.12999371$

Initialize

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

Use the Jacobian command in the Student MultivariateCalculus package to obtain the Jacobian

$\frac{\partial \left(x\,y\,z\right)}{\partial \left(u\,v\,w\right)}$ 

$\mathrm{Jacobian}\left(\left[x y\,x z\,y z\right]\,\left[x\,y\,z\right]\,\mathrm{output}\=\mathrm{determinant}\right)$ = $-2yxz$

Invert the transformation $u\=x y\,v\=x z\,w\=y z$ 

$S≔\mathrm{solve}\left(\left\{u\=x y\,v\=x z\,w\=y z\right\}\,\left\{x\,y\,z\right\}\,\mathrm{Explicit}\right)$

$\left\{x\=\frac{vu}{\sqrt{uvw}}\,y\=\frac{wu}{\sqrt{uvw}}\,z\=\frac{\sqrt{uvw}}{u}\right\}\,\left\{x\=-\frac{vu}{\sqrt{uvw}}\,y\=-\frac{wu}{\sqrt{uvw}}\,z\=-\frac{\sqrt{uvw}}{u}\right\}$

Use the eval and simplify commands to obtain the absolute value of the Jacobian $\frac{\partial \left(u\,v\,w\right)}{\partial \left(x\,y\,z\right)}$ 

$1\/ \mathrm{simplify}\left(\mathrm{eval}\left(2 x y z\,S_1\right)\right) assuming \mathrm{positive}$ = $\frac{1}{2\sqrt{u}\sqrt{v}\sqrt{w}}$

Use the MultiInt command from the Student MultivariateCalculus package to obtain the volume of $R$

(Note the use of the combine command to compact the products of radicals.)

$\mathrm{MultiInt}\left(\frac{1}{2\sqrt{u v w}}\,u\=1..5\,v\=4..15\,w\=10..20\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_{10}^{20}{\∫}_4^{15}{\∫}_1^5\frac{1}{2\sqrt{uvw}}\ⅆu\ⅆv\ⅆw$

$\mathrm{combine}\left(\mathrm{MultiInt}\left(\frac{1}{2\sqrt{u v w}}\,u\=1..5\,v\=4..15\,w\=10..20\right)\right)$

$-20\sqrt{30}\+20\sqrt{6}\+40\sqrt{2}-8\sqrt{10}\+40\sqrt{15}-40\sqrt{3}-80\+16\sqrt{5}$

$\mathrm{MultiInt}\left(\frac{1}{2\sqrt{u v w}}\,u\=1..5\,v\=4..15\,w\=10..20\,\mathrm{output}\=\mathrm{steps}\right)$

$\begin{array}{ccc}\multicolumn{3}{c}{{\int }_{10}^{20}{\int }_4^{15}{\int }_1^5\frac{1}{2\sqrt{uvw}}\ⅆu\ⅆv\ⅆw}\\ & \text{=}& {\int }_{10}^{20}{\int }_4^{15}\left(\genfrac{}{}{0ex}{}{\frac{u}{\sqrt{uvw}}}{\phantom{u=1..5}}|\genfrac{}{}{0ex}{}{\phantom{\frac{u}{\sqrt{uvw}}}}{u=1..5}\right)\ⅆv\ⅆw\hfill \\ & \text{=}& {\int }_{10}^{20}{\int }_4^{15}\frac{\sqrt{5}-1}{\sqrt{vw}}\ⅆv\ⅆw\hfill \\ & \text{=}& {\int }_{10}^{20}\left(\genfrac{}{}{0ex}{}{\frac{2v\left(\sqrt{5}-1\right)}{\sqrt{vw}}}{\phantom{v=4..15}}|\genfrac{}{}{0ex}{}{\phantom{\frac{2v\left(\sqrt{5}-1\right)}{\sqrt{vw}}}}{v=4..15}\right)\ⅆw\hfill \\ & \text{=}& {\int }_{10}^{20}-\frac{2\left(-5\sqrt{3}+\sqrt{5}\sqrt{3}+2\sqrt{5}-2\right)}{\sqrt{w}}\ⅆw\hfill \\ & \text{=}& \genfrac{}{}{0ex}{}{-4\sqrt{w}\left(-5\sqrt{3}+\sqrt{5}\sqrt{3}+2\sqrt{5}-2\right)}{\phantom{w=10..20}}|\genfrac{}{}{0ex}{}{\phantom{-4\sqrt{w}\left(-5\sqrt{3}+\sqrt{5}\sqrt{3}+2\sqrt{5}-2\right)}}{w=10..20}\hfill \end{array}$