Solve $3 x^2\+5 y^2\+7 z^2\=1$ for $z\=z\left(x\,y\right)$ 

$3 x^2\+5 y^2\+7 z^2\=1$$\stackrel{\text{assign to a name}}{\to }$$q$

$q$

$3x^2\+5y^2\+7z^2\=1$

$\stackrel{\text{solutions for z}}{\to }$

$\frac{1}{7}\sqrt{-21x^2-35y^2\+7}\,-\frac{1}{7}\sqrt{-21x^2-35y^2\+7}$

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

$Z$

Tools≻Tasks≻Browse:

Calculus - Vector≻Integration≻Surface Integration≻Surface Defined over a Rectangle

Table 6.3.10(a)   Task template for surface integration over a rectangle

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 numerically

${\int }_0^{3\/10}{\int }_0^{1\/10}x y \mathrm{\λ} \mathit{\ⅆ}y \mathit{\ⅆ}x$

${\∫}_0^{\frac{3}{10}}{\∫}_0^{\frac{1}{10}}xy\sqrt{1\+\frac{9x^2}{-21x^2-35y^2\+7}\+\frac{25y^2}{-21x^2-35y^2\+7}}\ⅆy\ⅆx$

$\stackrel{\text{at 10 digits}}{\to }$

$0.0002353725944$

Table 6.3.10(b)   Solution from first principles

Initialize

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

$$\begin{gathered} q≔{\left(x-1\/6\right)}^2\+{\left(y-1\/5\right)}^2\=1\/25\: \\ Z≔\mathrm{solve}\left(3 x^2\+5 y^2\+7 z^2\=1\,z\right)\: \end{gathered}$$

$\mathrm{\lambda }≔\mathrm{simplify}\left(\sqrt{1\+\mathrm{diff}{\left(Z_1\,x\right)}^2\+\mathrm{diff}{\left(Z_1\,y\right)}^2}\right)$

$\frac{1}{7}\sqrt{7}\sqrt{\frac{12x^2\+10y^2-7}{3x^2\+5y^2-1}}$

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

Evaluate the integral with the evalf command

$S≔\mathrm{MultiInt}\left(x y \mathrm{\λ}\,\left[x\,y\right]\=\mathrm{Rectangle}\left(0..3\/10\,0..1\/10\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^{\frac{1}{10}}{\∫}_0^{\frac{3}{10}}\frac{1}{7}xy\sqrt{7}\sqrt{\frac{12x^2\+10y^2-7}{3x^2\+5y^2-1}}\ⅆx\ⅆy$

$\mathrm{evalf}\left(S\right)$ = $0.0002353725944$

Use the SurfaceInt command from the Student VectorCalculus package

$Q≔\mathrm{Student}:-\mathrm{VectorCalculus}:-\mathrm{SurfaceInt}\left(x y\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,Z_1⟩\,\left[x\,y\right]\=\mathrm{Rectangle}\left(0..3\/10\,0..1\/10\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^{\frac{1}{10}}{\∫}_0^{\frac{3}{10}}xy\sqrt{1\+\frac{9x^2}{-21x^2-35y^2\+7}\+\frac{25y^2}{-21x^2-35y^2\+7}}\ⅆx\ⅆy$

$\mathrm{evalf}\left(Q\right)$ = $0.0002353725944$

$\mathrm{Int}\left(x y \mathrm{\λ}\,\left[y\=0..1\/10\,x\=0..3\/10\right]\right)\=\mathrm{evalf}\left(\mathrm{Int}\left(x y \mathrm{\lambda }\,\left[y\=0..1\/10\,x\=0..3\/10\right]\right)\right)$

${\∫}_0^{\frac{3}{10}}{\∫}_0^{\frac{1}{10}}\frac{1}{7}xy\sqrt{7}\sqrt{\frac{12x^2\+10y^2-7}{3x^2\+5y^2-1}}\ⅆy\ⅆx\=0.0002353725944$