$F\=x y \left(x^2\+y^2\right) \sqrt{1\+4 x^2\+4 y^2}$

so the surface integral itself is given by the sum

${\int }_1^3{\int }_{Y_{\mathrm{AB}}}^{Y_{\mathrm{CA}}}F \mathrm{dy} \mathrm{dx}\+{\int }_3^5{\int }_{Y_{\mathrm{AB}}}^{Y_{\mathrm{BC}}}F \mathrm{dy} \mathrm{dx}$ 

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

Table 9.6.8(a)   Solution by task template

Initialize

Loading Student:-Precalculus

Obtain the equations of the edges of the triangle

$\left[1\,2\right]\,\left[5\,1\right]$$\stackrel{\text{equation of line}}{\to }$$y\=-\frac{1}{4}x\+\frac{9}{4}$$\stackrel{\text{right hand side}}{\to }$$-\frac{1}{4}x\+\frac{9}{4}$$\stackrel{\text{assign to a name}}{\to }$$Y_{\mathrm{AB}}$

$\left[5\,1\right]\,\left[3\,3\right]$$\stackrel{\text{equation of line}}{\to }$$y\=-x\+6$$\stackrel{\text{right hand side}}{\to }$$-x\+6$$\stackrel{\text{assign to a name}}{\to }$$Y_{\mathrm{BC}}$

$\left[3\,3\right]\,\left[1\,2\right]$$\stackrel{\text{equation of line}}{\to }$$y\=\frac{1}{2}x\+\frac{3}{2}$$\stackrel{\text{right hand side}}{\to }$$\frac{1}{2}x\+\frac{3}{2}$$\stackrel{\text{assign to a name}}{\to }$$Y_{\mathrm{CA}}$

Write the integrand of the surface integral

$x y \left(x^2\+y^2\right) \sqrt{1\+4 x^2\+4 y^2}$$\stackrel{\text{assign to a name}}{\to }$$F$

Write and evaluate the surface integral

${\int }_1^3{\int }_{Y_{\mathrm{AB}}}^{Y_{\mathrm{CA}}}F \ⅆy \ⅆx\+{\int }_3^5{\int }_{Y_{\mathrm{AB}}}^{Y_{\mathrm{BC}}}F \ⅆy \ⅆx$

$\frac{1146089637}{417605000}\sqrt{21}\+\frac{321284403}{181741696}\sqrt{17}\ln \left(4\sqrt{341}\+\sqrt{17}\sqrt{21}\sqrt{341}\right)\+\frac{156333}{800000}\sqrt{5}\ln \left(8\sqrt{41}\+\sqrt{5}\sqrt{21}\sqrt{41}\right)-\frac{55305708887}{1122522240}\sqrt{46}-\frac{321284403}{181741696}\sqrt{17}\ln \left(21\sqrt{341}\+\sqrt{17}\sqrt{46}\sqrt{341}\right)\+\frac{13148711}{560000}\sqrt{73}-\frac{156333}{800000}\sqrt{5}\ln \left(18\sqrt{41}\+\sqrt{5}\sqrt{41}\sqrt{73}\right)\+\frac{55305708887}{1122522240}\sqrt{2}\sqrt{23}\+\frac{321284403}{181741696}\sqrt{17}\ln \left(21\sqrt{341}\+\sqrt{17}\sqrt{2}\sqrt{23}\sqrt{341}\right)\+\frac{892067025}{10690688}\sqrt{105}-\frac{321284403}{181741696}\sqrt{17}\ln \left(38\sqrt{341}\+\sqrt{17}\sqrt{105}\sqrt{341}\right)-\frac{1135077}{1024}\sqrt{2}\ln \left(73\right)\+\frac{1135077}{1024}\sqrt{2}\ln \left(8\sqrt{73}\sqrt{2}\+2\sqrt{73}\sqrt{105}\right)-\frac{1135077}{1024}\sqrt{2}\ln \left(2\right)$

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

$2032.782670$

Table 9.6.8(b)   Solution from first principles

Initialize

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

Implement the SurfaceInt command with the Triangle option

$\mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Triangle}\left(⟨1\,2⟩\,⟨5\,1⟩\,⟨3\,3⟩\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

$-\left({\∫}_1^3{\∫}_{\frac{3}{2}\+\frac{1}{2}x}^{\frac{9}{4}-\frac{1}{4}x}xy\left(x^2\+y^2\right)\sqrt{4x^2\+4y^2\+1}\ⅆy\ⅆx\right)-\left({\∫}_3^5{\∫}_{6-x}^{\frac{9}{4}-\frac{1}{4}x}xy\left(x^2\+y^2\right)\sqrt{4x^2\+4y^2\+1}\ⅆy\ⅆx\right)$

$q≔\mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Triangle}\left(⟨1\,2⟩\,⟨5\,1⟩\,⟨3\,3⟩\right)\right)\right)$

$\frac{1146089637}{417605000}\sqrt{21}\+\frac{156333}{800000}\sqrt{5}\ln \left(8\sqrt{41}\+\sqrt{5}\sqrt{21}\sqrt{41}\right)\+\frac{321284403}{181741696}\sqrt{17}\ln \left(4\sqrt{341}\+\sqrt{17}\sqrt{21}\sqrt{341}\right)\+\frac{13148711}{560000}\sqrt{73}-\frac{156333}{800000}\sqrt{5}\ln \left(18\sqrt{41}\+\sqrt{5}\sqrt{41}\sqrt{73}\right)-\frac{55305708887}{1122522240}\sqrt{46}-\frac{321284403}{181741696}\sqrt{17}\ln \left(21\sqrt{341}\+\sqrt{17}\sqrt{46}\sqrt{341}\right)\+\frac{55305708887}{1122522240}\sqrt{2}\sqrt{23}\+\frac{321284403}{181741696}\sqrt{17}\ln \left(21\sqrt{341}\+\sqrt{17}\sqrt{2}\sqrt{23}\sqrt{341}\right)\+\frac{892067025}{10690688}\sqrt{105}-\frac{1135077}{1024}\sqrt{2}\ln \left(73\right)-\frac{1135077}{1024}\sqrt{2}\ln \left(2\right)\+\frac{1135077}{1024}\sqrt{2}\ln \left(8\sqrt{73}\sqrt{2}\+2\sqrt{73}\sqrt{105}\right)-\frac{321284403}{181741696}\sqrt{17}\ln \left(38\sqrt{341}\+\sqrt{17}\sqrt{105}\sqrt{341}\right)$

$\mathrm{evalf}\left(q\right)$ = $2032.782670$$$

Table 9.6.8(c)   Solution via the SurfaceInt command

Initialize

$F≔x y \left(x^2\+y^2\right) \sqrt{1\+4 x^2\+4 y^2}\:$

Loading Student:-MultivariateCalculus

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

Obtain the equations of the edges of the triangle

$A\,B\,C≔\left[1\,2\right]\,\left[5\,1\right]\,\left[3\,3\right]\:$

Write and evaluate the surface integral

$\mathrm{MultiInt}\left(F\,y\=Y_{\mathrm{AB}}..Y_{\mathrm{CA}}\,x\=1..3\,\mathrm{output}\=\mathrm{integral}\right)\+\mathrm{MultiInt}\left(F\,y\=Y_{\mathrm{AB}}..Y_{\mathrm{BC}}\,x\=3..5\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_1^3{\∫}_{-\frac{1}{4}x\+\frac{9}{4}}^{\frac{1}{2}x\+\frac{3}{2}}xy\left(x^2\+y^2\right)\sqrt{4x^2\+4y^2\+1}\ⅆy\ⅆx\+{\∫}_3^5{\∫}_{-\frac{1}{4}x\+\frac{9}{4}}^{-x\+6}xy\left(x^2\+y^2\right)\sqrt{4x^2\+4y^2\+1}\ⅆy\ⅆx$

$\mathrm{MultiInt}\left(F\,y\=Y_{\mathrm{AB}}..Y_{\mathrm{CA}}\,x\=1..3\right)\+\mathrm{MultiInt}\left(F\,y\=Y_{\mathrm{AB}}..Y_{\mathrm{BC}}\,x\=3..5\right)$

$\frac{1146089637}{417605000}\sqrt{21}\+\frac{156333}{800000}\sqrt{5}\ln \left(8\sqrt{41}\+\sqrt{5}\sqrt{21}\sqrt{41}\right)\+\frac{321284403}{181741696}\sqrt{17}\ln \left(4\sqrt{341}\+\sqrt{17}\sqrt{21}\sqrt{341}\right)\+\frac{13148711}{560000}\sqrt{73}-\frac{156333}{800000}\sqrt{5}\ln \left(18\sqrt{41}\+\sqrt{5}\sqrt{41}\sqrt{73}\right)-\frac{55305708887}{1122522240}\sqrt{46}-\frac{321284403}{181741696}\sqrt{17}\ln \left(21\sqrt{341}\+\sqrt{17}\sqrt{46}\sqrt{341}\right)\+\frac{55305708887}{1122522240}\sqrt{2}\sqrt{23}\+\frac{321284403}{181741696}\sqrt{17}\ln \left(21\sqrt{341}\+\sqrt{17}\sqrt{2}\sqrt{23}\sqrt{341}\right)\+\frac{892067025}{10690688}\sqrt{105}-\frac{1135077}{1024}\sqrt{2}\ln \left(73\right)-\frac{1135077}{1024}\sqrt{2}\ln \left(2\right)\+\frac{1135077}{1024}\sqrt{2}\ln \left(8\sqrt{2}\sqrt{73}\+2\sqrt{105}\sqrt{73}\right)-\frac{321284403}{181741696}\sqrt{17}\ln \left(38\sqrt{341}\+\sqrt{17}\sqrt{105}\sqrt{341}\right)$

$\mathrm{MultiInt}\left(F\,y\=Y_{\mathrm{AB}}..Y_{\mathrm{CA}}\,x\=1..3.0\right)\+\mathrm{MultiInt}\left(F\,y\=Y_{\mathrm{AB}}..Y_{\mathrm{BC}}\,x\=3..5.0\right)$

Table 9.6.8(d)   Solution from first principles