Initialize

$F\=2-{\left(y-1\/2\right)}^2$$\stackrel{\text{assign}}{\to }$

Solve  $\cos \left(x\right)\=x$ for $Q$ 

$\cos \left(x\right)\=x$$\stackrel{\text{solve}}{\to }$$0.7390851332$$\stackrel{\text{assign to a name}}{\to }$$Q$

Iterate in the order $\mathrm{dx} \mathrm{dy}$ via the template in the Calculus palette

$\frac{{\int }_Q^1{\int }_{\arccos \left(y\right)}^{y}F \mathit{\ⅆ}x \mathit{\ⅆ}y}{{\int }_Q^1{\int }_{\arccos \left(y\right)}^{y}1 \mathit{\ⅆ}x \mathit{\ⅆ}y}$ = $1.821302486$

Solve the equation $\cos \left(x\right)\=x$ for $Q$ 

$Q≔\mathrm{fsolve}\left(x\=\cos \left(x\right)\,x\right)$

$0.7390851332$

Obtain the average value of $F$ 

$\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{FunctionAverage}\left(2-{\left(y-1\/2\right)}^2\,x\=\arccos \left(y\right)..y\,y\=Q..1\,\mathrm{output}\=\mathrm{integral}\right)$

$\frac{{\∫}_{0.7390851332}^1{\∫}_{\arccos \left(y\right)}^y\left(2-{\left(y-\frac{1}{2}\right)}^2\right)\ⅆx\ⅆy}{{\∫}_{0.7390851332}^1{\∫}_{\arccos \left(y\right)}^{y}1\ⅆx\ⅆy}$

$\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{FunctionAverage}\left(2-{\left(y-1\/2\right)}^2\,x\=\arccos \left(y\right)..y\,y\=Q..1\right)$

$1.821302486$