$z_{\pm }\= \pm \sqrt{7-21 x^2-35y^2}\/7$ 

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≻Multiple Integration≻2-D≻Over an Ellipse

Table 6.2.9(a)   Task template for integration over an ellipse

$r$

$\=\frac{1}{\sqrt{15}}\sqrt{\frac{{\tan }^2\left(\mathrm{\theta }\right)\+1}{\frac{1}{5}\+\frac{1}{3}{\tan }^2\left(\mathrm{\theta }\right)}}$

$\=\sqrt{\frac{{\sec }^2\left(\mathrm{\θ}\right)}{3\+5 {\tan }^2\left(\mathrm{\θ}\right)}}$$$

$\=\sqrt{\frac{1}{3 {\cos }^2\left(\mathrm{\θ}\right)\+5 {\sin }^2\left(\mathrm{\θ}\right)}}$

$\=\frac{1}{\sqrt{3\left(1-{\sin }^2\left(\mathrm{\theta }\right)\right)\+5 {\sin }^2\left(\mathrm{\theta }\right)}}$

$\=\frac{1}{\sqrt{3\+2 {\sin }^2\left(\mathrm{\θ}\right)}}$

$\genfrac{}{}{0ex}{}{3 x^2\+5 y^2\=1}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=r \cos \left(\mathrm{\theta }\right)\,y\=r \sin \left(\mathrm{\theta }\right)}$

$3r^2{\cos \left(\mathrm{\θ}\right)}^2\+5r^2{\sin \left(\mathrm{\θ}\right)}^2\=1$

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

$\frac{1}{\sqrt{2{\sin \left(\mathrm{\θ}\right)}^2\+3}}\,-\frac{1}{\sqrt{2{\sin \left(\mathrm{\θ}\right)}^2\+3}}$

Solve for $y\=y\left(x\right)$ 

$\genfrac{}{}{0ex}{}{q}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{z\=0}$ = $3x^2\+5y^2\=1$$\stackrel{\text{solutions for y}}{\to }$$\frac{1}{5}\sqrt{-15x^2\+5}\,-\frac{1}{5}\sqrt{-15x^2\+5}$$\stackrel{\text{assign to a name}}{\to }$$Y$

Write an appropriate iterated integral and evaluate

${\int }_{-\frac{1}{\sqrt{3}}}^{\frac{1}{\sqrt{3}}}{\int }_{Y_2}^{Y_1}\left(Z_1-Z_2\right) \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $\frac{4}{315}\sqrt{7}\sqrt{5}\mathrm{\π}\sqrt{3}$$$$\stackrel{\text{combine}}{\=}$$\frac{4}{315}\mathrm{\π}\sqrt{105}$$\stackrel{\text{at 10 digits}}{\to }$$0.4087840668$

Table 6.2.9(b)   Solution from first principles

Loading Student:-MultivariateCalculus

Access the MultiInt command via the Context Panel

$Z_1-Z_2$

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

$\stackrel{\text{MultiInt}}{\to }$

${\∫}_{-\frac{1}{3}\sqrt{3}}^{\frac{1}{3}\sqrt{3}}{\∫}_{-\frac{1}{5}\sqrt{-15x^2\+5}}^{\frac{1}{5}\sqrt{-15x^2\+5}}\frac{2}{7}\sqrt{-21x^2-35y^2\+7}\ⅆy\ⅆx$

$\=$

$\frac{4}{315}\sqrt{7}\sqrt{5}\mathrm{\π}\sqrt{3}$

$\stackrel{\text{combine}}{\=}$

$\frac{4}{315}\mathrm{\π}\sqrt{105}$

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

$0.4087840668$

Table 6.2.9(c)   Solution via the MultiInt command accessed through the Context Panel

Initialize

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

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

Top-level, using the Int and int commands

$\mathrm{Int}\left(Z_1-Z_2\,\left[y\=Y_2..Y_1\,x\=X_2..X_1\right]\right)\=\mathrm{int}\left(Z_1-Z_2\,\left[y\=Y_2..Y_1\,x\=X_2..X_1\right]\right)\mathit{ }$

${\∫}_{-\frac{1}{3}\sqrt{3}}^{\frac{1}{3}\sqrt{3}}{\∫}_{-\frac{1}{5}\sqrt{-15x^2\+5}}^{\frac{1}{5}\sqrt{-15x^2\+5}}\frac{2}{7}\sqrt{-21x^2-35y^2\+7}\ⅆy\ⅆx\=\frac{4}{315}\sqrt{7}\sqrt{5}\mathrm{\π}\sqrt{3}$

Use the MultiInt command from the Student MultivariateCalculus package

$\mathrm{MultiInt}\left(Z_1-Z_2\,y\=Y_2..Y_1\,x\=X_2..X_1\right)$ = $\frac{4}{315}\sqrt{7}\sqrt{5}\mathrm{\π}\sqrt{3}$$$

$\mathrm{MultiInt}\left(Z_1-Z_2\,y\=Y_2..Y_1\,x\=X_2..X_1\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_{-\frac{1}{3}\sqrt{3}}^{\frac{1}{3}\sqrt{3}}{\∫}_{-\frac{1}{5}\sqrt{-15x^2\+5}}^{\frac{1}{5}\sqrt{-15x^2\+5}}\frac{2}{7}\sqrt{-21x^2-35y^2\+7}\ⅆy\ⅆx$

Use the MultiInt command with a pre-defined domain option

$\mathrm{MultiInt}\left(Z_1-Z_2\,\left[x\,y\right]\=\mathrm{Region}\left(X_2..X_1\,Y_2..Y_1\right)\right)$ = $\frac{4}{315}\sqrt{7}\sqrt{5}\mathrm{\π}\sqrt{3}$$$

$\mathrm{MultiInt}\left(Z_1-Z_2\,\left[x\,y\right]\=\mathrm{Region}\left(X_2..X_1\,Y_2..Y_1\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_{-\frac{1}{3}\sqrt{3}}^{\frac{1}{3}\sqrt{3}}{\∫}_{-\frac{1}{5}\sqrt{-15x^2\+5}}^{\frac{1}{5}\sqrt{-15x^2\+5}}\frac{2}{7}\sqrt{-21x^2-35y^2\+7}\ⅆy\ⅆx$