On the faces $x\=1$ and $x\=2$, $\mathrm{d\σ}\=\mathrm{dy} \mathrm{dz}$.

On the faces $y\=3$ and $y\=4$, $\mathrm{d\σ}\=\mathrm{dx} \mathrm{dz}$.

On the faces $z\=5$ and $z\=6$, $\mathrm{d\σ}\=\mathrm{dx} \mathrm{dy}$.

On these faces,

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Calculus - Vector≻Integration≻Surface Integration≻Over a Box

Table 9.6.1(a)   Solution via task template

Define the function $f$ 

$f\left(x\,y\,z\right)\=x y z$$\stackrel{\text{assign as function}}{\to }$$f$

Write and evaluate the relevant surface integrals

${\int }_5^6{\int }_3^4\left(f\left(1\,y\,z\right)\+f\left(2\,y\,z\right) \right)\ⅆy \mathit{\ⅆ}z\+{\int }_5^6{\int }_1^2\left(f\left(x\,3\,z\right)\+f\left(x\,4\,z\right)\right) \mathit{\ⅆ}x \mathit{\ⅆ}z\+{\int }_3^4{\int }_1^2\left(f\left(x\,y\,5\right)\+f\left(x\,y\,6\right)\right) \mathit{\ⅆ}x \mathit{\ⅆ}y$

${\∫}_5^6{\∫}_3^{4}3yz\ⅆy\ⅆz\+{\∫}_5^6{\∫}_1^{2}7xz\ⅆx\ⅆz\+{\∫}_3^4{\∫}_1^{2}11xy\ⅆx\ⅆy$

$\=$

$\frac{693}{4}$

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

$\mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Box}\left(1..2\,3..4\,5..6\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_5^6{\∫}_3^{4}3st\ⅆs\ⅆt\+{\∫}_5^6{\∫}_1^{2}7st\ⅆs\ⅆt\+{\∫}_3^4{\∫}_1^{2}11st\ⅆs\ⅆt$

$\mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Box}\left(1..2\,3..4\,5..6\right)\right)$ = $\frac{693}{4}$$$