Initialize

$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$

Tools≻Task≻Browse:

Calculus - Vector≻Integration≻Surface Integration≻Surface Defined over a 2-D Region

Table 6.3.7(a)   Surface area via task template

Obtain $\mathrm{\lambda }\=\sqrt{1\+{\left(-2y\+1\right)}^2}$ 

$\sqrt{1\+{\left(\frac{\partial }{\partial x} F\right)}^2\+{\left(\frac{\partial }{\partial y} F\right)}^2}$ = $\sqrt{1\+{\left(-2y\+1\right)}^2}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{\λ}$

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

${\int }_Q^1{\int }_{\arccos \left(y\right)}^y\mathrm{\λ} \mathit{\ⅆ}x \mathit{\ⅆ}y$ = $0.1300942772$

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

${\int }_0^Q{\int }_{\cos \left(x\right)}^1\mathrm{\λ} \mathit{\ⅆ}y \mathit{\ⅆ}x\+{\int }_Q^1{\int }_x^1\mathrm{\λ} \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $0.1300942771$

Initialize

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

  $F≔2-{\left(y-1\/2\right)}^2\:$

Calculate the value of $x\=Q$, the solution of $\cos \left(x\right)\=x$ 

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

$0.7390851332$

For a solution from first principles, obtain $\mathrm{\λ}\=\sqrt{1\+F_x^2\+F_y^2}$ 

$\mathrm{\λ}≔\mathrm{sqrt}\left(1\+\mathrm{diff}{\left(F\,x\right)}^2\+\mathrm{diff}{\left(F\,y\right)}^2\right)$

$\sqrt{1\+{\left(-2y\+1\right)}^2}$

Top-level, using the Int and int commands

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

${\∫}_{0.7390851332}^1{\∫}_{\arccos \left(y\right)}^y\sqrt{1\+{\left(-2y\+1\right)}^2}\ⅆx\ⅆy\=0.1300942772$

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

${\∫}_0^{0.7390851332}{\∫}_{\cos \left(x\right)}^1\sqrt{1\+{\left(-2y\+1\right)}^2}\ⅆy\ⅆx\+{\∫}_{0.7390851332}^1{\∫}_x^1\sqrt{1\+{\left(-2y\+1\right)}^2}\ⅆy\ⅆx\=0.1300942772$

Use the MultiInt command from the Student MultivariateCalculus package

$\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{MultiInt}\left(\mathrm{\λ}\,x\=\arccos \left(y\right)..y\,y\=Q..1\right)$ = $0.1300942772$

$\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{MultiInt}\left(\mathrm{\λ}\,x\=\arccos \left(y\right)..y\,y\=Q..1\,\mathrm{output}\=\mathrm{integral}\right)$

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

$\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{MultiInt}\left(\mathrm{\λ}\,x\=\arccos \left(y\right)..y\,y\=Q..1\,\mathrm{output}\=\mathrm{steps}\right)$

$\begin{array}{ccc}\multicolumn{3}{c}{{\int }_{0.7390851332}^1{\int }_{\arccos \left(y\right)}^y\sqrt{1+{\left(-2y+1\right)}^2}\ⅆx\ⅆy}\\ & \text{=}& {\int }_{0.7390851332}^1\left(\genfrac{}{}{0ex}{}{\sqrt{1+{\left(-2y+1\right)}^2}x}{\phantom{x=\arccos \left(y\right)..y}}|\genfrac{}{}{0ex}{}{\phantom{\sqrt{1+{\left(-2y+1\right)}^2}x}}{x=\arccos \left(y\right)..y}\right)\ⅆy\hfill \\ & \text{=}& {\int }_{0.7390851332}^1\sqrt{1+{\left(-2y+1\right)}^2}\left(y-\arccos \left(y\right)\right)\ⅆy\hfill \\ & \text{=}& \genfrac{}{}{0ex}{}{\int \sqrt{1+{\left(-2y+1\right)}^2}\left(y-\arccos \left(y\right)\right)\ⅆy}{\phantom{y=0.7390851332..1}}|\genfrac{}{}{0ex}{}{\phantom{\int \sqrt{1+{\left(-2y+1\right)}^2}\left(y-\arccos \left(y\right)\right)\ⅆy}}{y=0.7390851332..1}\hfill \end{array}$

Use the SurfaceInt command from the Student VectorCalculus package

$\mathrm{Student}:-\mathrm{VectorCalculus}:-\mathrm{SurfaceInt}\left(1\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,F⟩\,\left[y\,x\right]\=\mathrm{Region}\left(Q..1\, \arccos \left(y\right)..y\right)\right)\right)$

$0.1300942772$