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 an Ellipse

Table 6.3.11(a)   Task template for surface integration over an ellipse

Solve the equation of the ellipse for $y\=y\left(x\right)$ 

$3 x^2\+5 y^2\=1\/2$$\stackrel{\text{solutions for y}}{\to }$$\frac{1}{10}\sqrt{-60x^2\+10}\,-\frac{1}{10}\sqrt{-60x^2\+10}$$\stackrel{\text{assign to a name}}{\to }$$Y$

Obtain $x\=x_L$ and $x\=x_R$ for the ellipse

$Y_1\=0$

$\frac{1}{10}\sqrt{-60x^2\+10}\=0$

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

$-\frac{1}{6}\sqrt{6}\,\frac{1}{6}\sqrt{6}$

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

${\int }_{-1\/\sqrt{6}}^{1\/\sqrt{6}}{\int }_{Y_2}^{Y_1}{\left(x y\right)}^2 \mathrm{\λ} \mathit{\ⅆ}y \mathit{\ⅆ}x$

${\∫}_{-\frac{1}{6}\sqrt{6}}^{\frac{1}{6}\sqrt{6}}{\∫}_{-\frac{1}{10}\sqrt{-60x^2\+10}}^{\frac{1}{10}\sqrt{-60x^2\+10}}{x}^2{y}^2\sqrt{1\+\frac{9x^2}{-21x^2-35y^2\+7}\+\frac{25y^2}{-21x^2-35y^2\+7}}\ⅆy\ⅆx$

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

$0.0003283936763$

Table 6.3.11(b)   Solution from first principles

Initialize

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

$$\begin{gathered} q≔3 x^2\+5 y^2\=1\/2\: \\ Z≔\mathrm{solve}\left(3 x^2\+5 y^2\+7 z^2\=1\,z\right)\: \\ Y≔\mathrm{solve}\left(q\,y\right)\: \\ X≔\mathrm{solve}\left(\mathrm{eval}\left(q\,y\=0\right)\, x\right)\: \end{gathered}$$

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

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

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

Evaluate the integral with the evalf command

$S≔\mathrm{MultiInt}\left({\left(x y\right)}^2\cdot \mathrm{\λ}\,\left[x\,y\right]\=\mathrm{Ellipse}\left(q\,\left[r\,\mathrm{\θ}\right]\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^{\frac{1}{60}\sqrt{6}\sqrt{10}\sqrt{\frac{{\tan \left(\mathrm{\θ}\right)}^2\+1}{\frac{1}{10}\+\frac{1}{6}{\tan \left(\mathrm{\θ}\right)}^2}}}\frac{1}{7}{r}^5{\cos \left(\mathrm{\θ}\right)}^2{\sin \left(\mathrm{\θ}\right)}^2\sqrt{7}\sqrt{\frac{12r^2{\cos \left(\mathrm{\θ}\right)}^2\+10r^2{\sin \left(\mathrm{\θ}\right)}^2-7}{3r^2{\cos \left(\mathrm{\θ}\right)}^2\+5r^2{\sin \left(\mathrm{\θ}\right)}^2-1}}\ⅆr\ⅆ\mathrm{\θ}$

$\mathrm{evalf}\left(S\right)$ = $0.0003283936762$

Use the SurfaceInt command from the Student VectorCalculus package

$Q≔\mathrm{Student}:-\mathrm{VectorCalculus}:-\mathrm{SurfaceInt}\left({\left(x y\right)}^2\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,Z_1⟩\,\left[x\,y\right]\=\mathrm{Ellipse}\left(q\,\left[r\,\mathrm{\θ}\right]\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^{\frac{1}{60}\sqrt{6}\sqrt{10}\sqrt{\frac{{\tan \left(\mathrm{\θ}\right)}^2\+1}{\frac{1}{10}\+\frac{1}{6}{\tan \left(\mathrm{\θ}\right)}^2}}}{r}^5{\cos \left(\mathrm{\θ}\right)}^2{\sin \left(\mathrm{\θ}\right)}^2\sqrt{1\+\frac{9r^2{\cos \left(\mathrm{\θ}\right)}^2}{-21r^2{\cos \left(\mathrm{\θ}\right)}^2-35r^2{\sin \left(\mathrm{\θ}\right)}^2\+7}\+\frac{25r^2{\sin \left(\mathrm{\θ}\right)}^2}{-21r^2{\cos \left(\mathrm{\θ}\right)}^2-35r^2{\sin \left(\mathrm{\θ}\right)}^2\+7}}\ⅆr\ⅆ\mathrm{\θ}$

$\mathrm{evalf}\left(Q\right)$ = $0.0003283936762$

$\mathrm{Int}\left({\left(x y\right)}^2 \mathrm{\λ}\,\left[y\=Y_2..Y_1\,x\=X_2..X_1\right]\right)\=\mathrm{evalf}\left(\mathrm{Int}\left({\left(x y\right)}^2 \mathrm{\λ}\,\left[y\=Y_2..Y_1\,x\=X_2..X_1\right]\right)\right)$

${\∫}_{-\frac{1}{6}\sqrt{6}}^{\frac{1}{6}\sqrt{6}}{\∫}_{-\frac{1}{10}\sqrt{-60x^2\+10}}^{\frac{1}{10}\sqrt{-60x^2\+10}}\frac{1}{7}{x}^2{y}^2\sqrt{7}\sqrt{\frac{12x^2\+10y^2-7}{3x^2\+5y^2-1}}\ⅆy\ⅆx\=0.0003283936763$