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 Disk

Table 6.3.9(a)   Task template for surface integration over a disk

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

${\left(x-1\/6\right)}^2\+{\left(y-1\/5\right)}^2\=1\/5^2$$\stackrel{\text{solutions for y}}{\to }$$\frac{1}{5}\+\frac{1}{30}\sqrt{-900x^2\+300x\+11}\,\frac{1}{5}-\frac{1}{30}\sqrt{-900x^2\+300x\+11}$$\stackrel{\text{assign to a name}}{\to }$$Y$

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

$Y_1\=1\/5$

$\frac{1}{5}\+\frac{1}{30}\sqrt{-900x^2\+300x\+11}\=\frac{1}{5}$

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

$-\frac{1}{30}\,\frac{11}{30}$

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

${\int }_{-1\/30}^{11\/30}{\int }_{Y_2}^{Y_1}x y \mathrm{\λ} \mathit{\ⅆ}y \mathit{\ⅆ}x$

${\∫}_{-\frac{1}{30}}^{\frac{11}{30}}{\∫}_{\frac{1}{5}-\frac{1}{30}\sqrt{-900x^2\+300x\+11}}^{\frac{1}{5}\+\frac{1}{30}\sqrt{-900x^2\+300x\+11}}xy\sqrt{1\+\frac{9x^2}{-21x^2-35y^2\+7}\+\frac{25y^2}{-21x^2-35y^2\+7}}\ⅆy\ⅆx$

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

$0.005897390459$

Table 6.3.9(b)   Solution from first principles

Initialize

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

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

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

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

Use the MultiInt command with a pre-defined domain option

$S≔\mathrm{MultiInt}\left(x y \mathrm{\λ}\,\left[x\,y\right]\=\mathrm{Circle}\left(⟨1\/6\,1\/5⟩\,1\/5\,\left[r\,\mathrm{\θ}\right]\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^{\frac{1}{5}}{\∫}_0^{2\mathrm{\π}}\frac{1}{105}\sqrt{7}r\left(6r\cos \left(\mathrm{\θ}\right)\+1\right)\left(5r\sin \left(\mathrm{\θ}\right)\+1\right)\sqrt{-\frac{2\left(15r^2{\cos \left(\mathrm{\θ}\right)}^2\+30r\cos \left(\mathrm{\θ}\right)\+30r\sin \left(\mathrm{\θ}\right)\+75r^2-47\right)}{120r^2{\cos \left(\mathrm{\θ}\right)}^2-60r\cos \left(\mathrm{\θ}\right)-120r\sin \left(\mathrm{\θ}\right)-300r^2\+43}}\ⅆ\mathrm{\θ}\ⅆr$

$\mathrm{evalf}\left(S\right)$ = $0.005897390462$$$

Use the SurfaceInt command from the Student VectorCalculus package

$Q≔\mathrm{Student}:-\mathrm{VectorCalculus}:-\mathrm{SurfaceInt}\left(x y\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,Z_1⟩\,\left[x\,y\right]\=\mathrm{Circle}\left(⟨1\/6\,1\/5⟩\,1\/5\,\left[r\,\mathrm{\θ}\right]\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^{\frac{1}{5}}{\∫}_0^{2\mathrm{\π}}\frac{1}{105}r\sqrt{-\frac{210r^2{\cos \left(\mathrm{\θ}\right)}^2\+420r\cos \left(\mathrm{\θ}\right)\+420r\sin \left(\mathrm{\θ}\right)\+1050r^2-658}{120r^2{\cos \left(\mathrm{\θ}\right)}^2-60r\cos \left(\mathrm{\θ}\right)-120r\sin \left(\mathrm{\θ}\right)-300r^2\+43}}\left(30\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\θ}\right)r^2\+6r\cos \left(\mathrm{\θ}\right)\+5r\sin \left(\mathrm{\θ}\right)\+1\right)\ⅆ\mathrm{\θ}\ⅆr$

$\mathrm{evalf}\left(Q\right)$ = $0.005897390462$$$

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

${\∫}_{-\frac{1}{30}}^{\frac{11}{30}}{\∫}_{\frac{1}{5}-\frac{1}{30}\sqrt{-900x^2\+300x\+11}}^{\frac{1}{5}\+\frac{1}{30}\sqrt{-900x^2\+300x\+11}}\frac{1}{7}xy\sqrt{7}\sqrt{\frac{12x^2\+10y^2-7}{3x^2\+5y^2-1}}\ⅆy\ⅆx\=0.005897390459$