$\frac{{\int }_0^X{\int }_{\sin \left(x\right)}^{f\left(x\right)}F \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_X^{\mathrm{\pi }\/2}{\int }_{f\left(x\right)}^{\sin \left(x\right)}F \mathit{\ⅆ}y \mathit{\ⅆ}x}{{\int }_0^X{\int }_{\sin \left(x\right)}^{f\left(x\right)}1 \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_X^{\mathrm{\pi }\/2}{\int }_{f\left(x\right)}^{\sin \left(x\right)}1 \mathit{\ⅆ}y \mathit{\ⅆ}x}$

$\doteq \frac{1.572488959}{0.3016523782}$

$\doteq 5.212917493$

Initialize

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

$f\=\arctan \left(x\+1\right)-1\/2$$\stackrel{\text{assign}}{\to }$

$g\=\sin \left(x\right)$$\stackrel{\text{assign}}{\to }$

Obtain the intersections of the curves bounding $R$ 

$f\=g$

$\arctan \left(x\+1\right)-\frac{1}{2}\=\sin \left(x\right)$

$\stackrel{\text{solve}}{\to }$

$0.5059163080$

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

$X$

Obtain the average value of $F$ 

$\frac{{\int }_0^X{\int }_g^{f}F \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_X^{\mathrm{\π}\/2}{\int }_f^{g}F \mathit{\ⅆ}y \mathit{\ⅆ}x}{{\int }_0^X{\int }_g^{f}1 \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_X^{\mathrm{\pi }\/2}{\int }_f^{g}1 \mathit{\ⅆ}y \mathit{\ⅆ}x}$ = $5.212917493$

Initialize

$F≔3 x^2\+2 y^2\+1\:$

$f≔\arctan \left(x\+1\right)-1\/2\:$

$g≔\sin \left(x\right)\:$

$X≔\mathrm{fsolve}\left(f\=g\,x\right)\:$

$Q≔\frac{\mathrm{Int}\left(F\,\left[y\=g..f\,x\=0..X\right]\right)\+\mathrm{Int}\left(F\,\left[y\=f..g\,x\=X..\mathrm{\pi }\/2\right]\right)}{\mathrm{Int}\left(1\,\left[y\=g..f\,x\=0..X\right]\right)\+\mathrm{Int}\left(1\,\left[y\=f..g\,x\=X..\mathrm{\pi }\/2\right]\right)}$

$\frac{{\∫}_0^{0.5059163080}{\∫}_{\sin \left(x\right)}^{\arctan \left(x\+1\right)-\frac{1}{2}}\left(3x^2\+2y^2\+1\right)\ⅆy\ⅆx\+{\∫}_{0.5059163080}^{\frac{1}{2}\mathrm{\π}}{\∫}_{\arctan \left(x\+1\right)-\frac{1}{2}}^{\sin \left(x\right)}\left(3x^2\+2y^2\+1\right)\ⅆy\ⅆx}{{\∫}_0^{0.5059163080}{\∫}_{\sin \left(x\right)}^{\arctan \left(x\+1\right)-\frac{1}{2}}1\ⅆy\ⅆx\+{\∫}_{0.5059163080}^{\frac{1}{2}\mathrm{\π}}{\∫}_{\arctan \left(x\+1\right)-\frac{1}{2}}^{\sin \left(x\right)}1\ⅆy\ⅆx}$

$q≔\mathrm{value}\left(Q\right)$

$5.212917493$