Initialize

$F\=2 x\+3 y\+1$$\stackrel{\text{assign}}{\to }$

Obtain $\mathrm{d\σ}\prime \=\sqrt{1\+F_x^2\+F_y^2}$ 

$\sqrt{1\+{\left(\frac{\partial }{\partial x} F\right)}^2\+{\left(\frac{\partial }{\partial y} F\right)}^2}$ = $\sqrt{14}$$$

Write and evaluate the relevant iterated integral

${\int }_0^{\mathrm{\π}\/3}{\int }_{\sin \left(x\right)}^{\sin \left(2 x\right)}\sqrt{14}\mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_{\mathrm{\π}\/3}^{\mathrm{\π}}{\int }_{\sin \left(2 x\right)}^{\sin \left(x\right)}\sqrt{14}\mathit{\ⅆ}y \mathit{\ⅆ}x$ = $\frac{5}{2}\sqrt{14}$$$

Table 6.3.3(a)   Solution from first principles

Initialize

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

$F≔2 x\+3 y\+1\:$

Apply the SurfaceArea command from the Student MultivariateCalculus package

$$\begin{gathered} \mathrm{SurfaceArea}\left(F\,x\=0..\mathrm{\π}\/3\,y\=\sin \left(x\right)..\sin \left(2 x\right)\,\mathrm{output}\=\mathrm{integral}\right)\+ \\ \mathrm{SurfaceArea}\left(F\,x\= \mathrm{\π}\/3..\mathrm{\π}\,y\=\sin \left(2 x\right)..\sin \left(x\right)\,\mathrm{output}\=\mathrm{integral}\right) \end{gathered}$$

${\∫}_0^{\frac{1}{3}\mathrm{\π}}{\∫}_{\sin \left(x\right)}^{\sin \left(2x\right)}\sqrt{14}\ⅆy\ⅆx\+{\∫}_{\frac{1}{3}\mathrm{\π}}^{\mathrm{\π}}{\∫}_{\sin \left(2x\right)}^{\sin \left(x\right)}\sqrt{14}\ⅆy\ⅆx$

$$\begin{gathered} \mathrm{SurfaceArea}\left(F\,x\=0..\mathrm{\π}\/3\, y\=\sin \left(x\right)..\sin \left(2 x\right)\right)\+ \\ \mathrm{SurfaceArea}\left(F\,x\= \mathrm{\π}\/3..\mathrm{\π}\,y\=\sin \left(2 x\right)..\sin \left(x\right)\right) \end{gathered}$$

$\frac{5}{2}\sqrt{14}$

The top-level Int command returns the inert iterated integral

$\mathrm{Int}\left(\sqrt{14}\,\left[y\=\sin \left(x\right)..\sin \left(2 x\right)\,x\=0..\mathrm{\π}\/3\right]\right)\+\mathrm{Int}\left(\sqrt{14}\,\left[y\=\sin \left(2 x\right)..\sin \left(x\right)\,x\= \mathrm{\π}\/3..\mathrm{\π}\right]\right)$

${\∫}_0^{\frac{1}{3}\mathrm{\π}}{\∫}_{\sin \left(x\right)}^{\sin \left(2x\right)}\sqrt{14}\ⅆy\ⅆx\+{\∫}_{\frac{1}{3}\mathrm{\π}}^{\mathrm{\π}}{\∫}_{\sin \left(2x\right)}^{\sin \left(x\right)}\sqrt{14}\ⅆy\ⅆx$