Initialize

$z_1\=7-x^2-y^2$$\stackrel{\text{assign}}{\to }$$$

Tools≻Tasks≻Browse:

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

Table 6.3.13(a)   Solution by task template

Obtain $\mathrm{\λ}\=\sqrt{1\+f_x^2\+f_y^2}$

$\mathrm{\λ}\=\sqrt{1\+{\left(\frac{\partial }{\partial x} z_1\right)}^2\+{\left(\frac{\partial }{\partial y} z_1\right)}^2}$$\stackrel{\text{assign}}{\to }$$$

Write an appropriate iterated integral and evaluate

${\int }_0^1{\int }_x^{2-x^2}x y \mathrm{\λ} \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $-\frac{799}{960}-\frac{1}{4}\ln \left(2\right)\+\frac{1}{4}\ln \left(3\+\sqrt{17}\right)-\frac{1}{4}\ln \left(-3\+\sqrt{17}\right)\+\frac{29}{64}\sqrt{17}$$$

Table 6.3.13(b)   Solution from first principles

Initialize

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

$z_1≔7-x^2-y^2\:$

$\mathrm{\lambda }≔\sqrt{1\+\mathrm{diff}{\left(z_1\,x\right)}^2\+\mathrm{diff}{\left(z_1\,y\right)}^2}$

$\sqrt{4x^2\+4y^2\+1}$

Form the integral via the MultiInt command with a pre-defined domain option

Evaluate the exact answer with the evalf command

$\mathrm{MultiInt}\left(xy\mathrm{\λ}\,\left[x\,y\right]\=\mathrm{Region}\left(0..1\,x..2-x^2\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^1{\∫}_x^{-x^2\+2}xy\sqrt{4x^2\+4y^2\+1}\ⅆy\ⅆx$$$

$S≔\mathrm{MultiInt}\left(xy\mathrm{\lambda }\,\left[x\,y\right]\=\mathrm{Region}\left(0..1\,x..2-x^2\right)\right)$

$-\frac{799}{960}-\frac{1}{4}\ln \left(2\right)\+\frac{1}{4}\ln \left(3\+\sqrt{17}\right)-\frac{1}{4}\ln \left(-3\+\sqrt{17}\right)\+\frac{29}{64}\sqrt{17}$

$\mathrm{evalf}\left(S\right)$ = $1.324515296$$$

Use the SurfaceInt command from the Student VectorCalculus package

$\mathrm{Student}:-\mathrm{VectorCalculus}:-\mathrm{SurfaceInt}\left(x y\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,z_1⟩\,\left[x\,y\right]\=\mathrm{Region}\left(0..1\,x..2-x^2\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^1{\∫}_x^{-x^2\+2}xy\sqrt{4x^2\+4y^2\+1}\ⅆy\ⅆx$

$q≔\mathrm{Student}:-\mathrm{VectorCalculus}:-\mathrm{SurfaceInt}\left(x y\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,z_1⟩\,\left[x\,y\right]\=\mathrm{Region}\left(0..1\,x..2-x^2\right)\right)\right)$

$-\frac{799}{960}-\frac{1}{4}\ln \left(2\right)\+\frac{1}{4}\ln \left(3\+\sqrt{17}\right)-\frac{1}{4}\ln \left(-3\+\sqrt{17}\right)\+\frac{29}{64}\sqrt{17}$

$\mathrm{evalf}\left(q\right)$ = $1.324515296$$$

$\mathrm{Int}\left(x y \mathrm{\λ}\,\left[y\=x..2-x^2\,x\=0..1\right]\right)\=\mathrm{evalf}\left(\mathrm{Int}\left(x y \mathrm{\λ}\,\left[y\=x..2-x^2\,x\=0..1\right]\right)\right)$

${\∫}_0^1{\∫}_x^{-x^2\+2}xy\sqrt{4x^2\+4y^2\+1}\ⅆy\ⅆx\=1.324515295$