${\int }_Q^1{\int }_{\arccos \left(y\right)}^{y}F \mathrm{dx} \mathrm{dy}$ = $0.1812403381$ 

${\int }_0^Q{\int }_{\cos \left(x\right)}^{1}F \mathrm{dy} \mathrm{dx}\+{\int }_Q^1{\int }_x^{1}F \mathrm{dy} \mathrm{dx}$ =  $0.1812403381$

Initialize

Loading Student:-MultivariateCalculus

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

$X_L\=\arccos \left(y\right)$$\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$

Access the MultiInt command via the Context Panel

$F$ = $2-{\left(y-\frac{1}{2}\right)}^2$$\stackrel{\text{MultiInt}}{\to }$${\∫}_{0.7390851332}^1{\∫}_{\arccos \left(y\right)}^y\left(2-{\left(y-\frac{1}{2}\right)}^2\right)\ⅆx\ⅆy$$\=$$0.1812403381$$$

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

${\int }_Q^1{\int }_{\arccos \left(y\right)}^{y}F \mathit{\ⅆ}x \mathit{\ⅆ}y$ = $0.1812403381$$$

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

${\int }_0^Q{\int }_{\cos \left(x\right)}^{1}F \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_Q^1{\int }_x^{1}F \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $0.1812403381$$$

Tools≻Tasks≻Browse:

Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cartesian 2-D

Table 6.2.7(a)   Iteration in the order $\mathrm{dx} \mathrm{dy}$ via visualization task-template

Initialize

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

  $F≔2-{\left(y-1\/2\right)}^2\:$$$

Calculate the value of $Q$ 

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

$0.7390851332$

Top-level, using the Int and int commands

$\mathrm{Int}\left(F\,\left[x\=\arccos \left(y\right)..y\,y\=Q..1\right]\right)\=\mathrm{int}\left(F\,\left[x\=\arccos \left(y\right)..y\,y\=Q..1\right]\right)$

${\∫}_{0.7390851332}^1{\∫}_{\arccos \left(y\right)}^y\left(2-{\left(y-\frac{1}{2}\right)}^2\right)\ⅆx\ⅆy\=0.1812403381$

$$\begin{gathered} \mathrm{Int}\left(F\,\left[y\=\cos \left(x\right)..1\,x\=0..Q\right]\right)\+\mathrm{Int}\left(F\,\left[y\=x..1\,x\=Q..1\right]\right)\= \\ \mathrm{int}\left(F\,\left[y\=\cos \left(x\right)..1\,x\=0..Q\right]\right)\+\mathrm{int}\left(F\,\left[y\=x..1\,x\=Q..1\right]\right) \end{gathered}$$

${\∫}_0^{0.7390851332}{\∫}_{\cos \left(x\right)}^1\left(2-{\left(y-\frac{1}{2}\right)}^2\right)\ⅆy\ⅆx\+{\∫}_{0.7390851332}^1{\∫}_x^1\left(2-{\left(y-\frac{1}{2}\right)}^2\right)\ⅆy\ⅆx\=0.1812403381$

Use the MultiInt command from the Student MultivariateCalculus package

$\mathrm{MultiInt}\left(F\,x\=\arccos \left(y\right)..y\,y\=Q..1\right)$ = $0.1812403381$$$

$\mathrm{MultiInt}\left(F\,x\=\arccos \left(y\right)..y\,y\=Q..1\,\mathrm{output}\=\mathrm{integral}\right)$

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

$\mathrm{MultiInt}\left(F\,x\=\arccos \left(y\right)..y\,y\=Q..1\,\mathrm{output}\=\mathrm{steps}\right)$

$0.1812403381$

Use the MultiInt command with a pre-defined domain option

$\mathrm{MultiInt}\left(F\,\left[y\,x\right]\=\mathrm{Region}\left(Q..1\,\arccos \left(y\right)..y\right)\right)$ = $0.1812403381$

$\mathrm{MultiInt}\left(F\,\left[y\,x\right]\=\mathrm{Region}\left(Q..1\,\arccos \left(y\right)..y\right)\,\mathrm{output}\=\mathrm{integral}\right)$

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