${\int }_1^2{\int }_1^{3}x y \left(x^2\+y^2\right) \sqrt{1\+4 x^2\+4 y^2} \mathrm{dy} \mathrm{dx}$

$\=\frac{81}{112}-\frac{147}{40}\sqrt{21}-\frac{48749}{1680}\sqrt{41}\+\frac{53371}{840}\sqrt{53}$

$\doteq 260.64$

Tools≻Tasks≻Browse:
Calculus - Vector≻Integration≻Surface Integration≻Surface Defined over a Rectangle

Table 9.6.6(a)   Solution by task template

Initialize

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

Implement the SurfaceInt command

$\mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Rectangle}\left(1..2\,1..3\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_1^3{\∫}_1^{2}xy\left(x^2\+y^2\right)\sqrt{4x^2\+4y^2\+1}\ⅆx\ⅆy$

$\mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Rectangle}\left(1..2\,1..3\right)\right)\right)$

$\frac{81}{112}-\frac{147}{40}\sqrt{21}-\frac{48749}{1680}\sqrt{41}\+\frac{53371}{840}\sqrt{53}$

Table 9.6.6(b)   Solution via the SurfaceInt command