Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Surface Area

Table 6.3.8(a)   Solution by SurfaceArea command implemented in a task template

$F\=\sqrt{2-x^2-4 y^2}$$\stackrel{\text{assign}}{\to }$

$\sqrt{1\+{\left(\frac{\partial }{\partial x} F\right)}^2\+{\left(\frac{\partial }{\partial y} F\right)}^2}$

$\sqrt{1\+\frac{x^2}{-x^2-4y^2\+2}\+\frac{16y^2}{-x^2-4y^2\+2}}$

$\stackrel{\text{simplify}}{\=}$

$\sqrt{-\frac{2\left(6y^2\+1\right)}{x^2\+4y^2-2}}$

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

$f$

Table 6.3.8(b)   Calculation of $\mathrm{\lambda }\=\sqrt{1\+F_x^2\+F_y^2}$

Tools≻Tasks≻Browse:

Calculus - Vector≻Integration≻Multiple Integration≻2-D≻Over an Ellipse

Table 6.2.8(c)   Task template for integration over an ellipse

$r$

$\=\frac{1}{2}\sqrt{\frac{{\tan }^2\left(\mathrm{\theta }\right)\+1}{\frac{1}{4}\+{\tan }^2\left(\mathrm{\theta }\right)}}$

$\=\sqrt{\frac{{\sec }^2\left(\mathrm{\θ}\right)}{1\+4 {\tan }^2\left(\mathrm{\θ}\right)}}$$$

$\=\sqrt{\frac{1}{{\cos }^2\left(\mathrm{\θ}\right)\+4 {\sin }^2\left(\mathrm{\θ}\right)}}$

$\=\frac{1}{\sqrt{1-{\sin }^2\left(\mathrm{\theta }\right)\+4 {\sin }^2\left(\mathrm{\theta }\right)}}$

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

$\genfrac{}{}{0ex}{}{x^2\+4 y^2\=1}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=r \cos \left(\mathrm{\theta }\right)\,y\=r \sin \left(\mathrm{\theta }\right)}$

$r^2{\cos \left(\mathrm{\θ}\right)}^2\+4r^2{\sin \left(\mathrm{\θ}\right)}^2\=1$

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

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

Solve for $y\=y\left(x\right)$ 

$x^2\+4 y^2\=1$$\stackrel{\text{solutions for y}}{\to }$$\frac{1}{2}\sqrt{-x^2\+1}\,-\frac{1}{2}\sqrt{-x^2\+1}$$\stackrel{\text{assign to a name}}{\to }$$Y$

Write an appropriate iterated integral and evaluate

${\int }_{-1}^{1.}{\int }_{Y_2}^{Y_1}f \mathit{\ⅆ}y \mathit{\ⅆ}x$$\stackrel{\text{at 10 digits}}{\to }$$2.155517778$$$  

Table 6.2.8(d)   Solution from first principles

Initialize

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

$F≔\sqrt{2-x^2-4 y^2}\:$

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

$\sqrt{-\frac{2\left(6y^2\+1\right)}{x^2\+4y^2-2}}$

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

Evaluate numerically with the evalf command.

$q≔\mathrm{MultiInt}\left(\mathrm{\λ}\,\left[x\,y\right]\=\mathrm{Ellipse}\left(x^2\+4 y^2\=1\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^{\frac{1}{2}\sqrt{\frac{{\tan \left(y\right)}^2\+1}{\frac{1}{4}\+{\tan \left(y\right)}^2}}}\sqrt{-\frac{2\left(6x^2{\sin \left(y\right)}^2\+1\right)}{x^2{\cos \left(y\right)}^2\+4x^2{\sin \left(y\right)}^2-2}}x\ⅆx\ⅆy$

$\mathrm{evalf}\left(q\right)$ = $2.155517778$$$

Use the SurfaceInt command from the Student VectorCalculus package

$Q≔\mathrm{simplify}\left(\mathrm{Student}:-\mathrm{VectorCalculus}:-\mathrm{SurfaceInt}\left(1\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,F⟩\,\left[x\,y\right]\=\mathrm{Ellipse}\left(x^2\+4 y^2\=1\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^{\frac{1}{\sqrt{-3{\cos \left(y\right)}^2\+4}}}\sqrt{-\frac{12x^2{\cos \left(y\right)}^2-12x^2-2}{3x^2{\cos \left(y\right)}^2-4x^2\+2}}x\ⅆx\ⅆy$

$\mathrm{evalf}\left(Q\right)$ = $2.155517778$$$

Solution from first principles

$Y≔\mathrm{solve}\left(x^2\+4 y^2\=1\,y\right)$

$\frac{1}{2}\sqrt{-x^2\+1}\,-\frac{1}{2}\sqrt{-x^2\+1}$

Top-level, using the Int and evalf  commands

$q≔\mathrm{Int}\left(\mathrm{\λ}\,\left[y\=Y_2..Y_1\,x\=-1..1\right]\right)$

${\∫}_{-1}^1{\∫}_{-\frac{1}{2}\sqrt{-x^2\+1}}^{\frac{1}{2}\sqrt{-x^2\+1}}\sqrt{-\frac{12y^2\+2}{x^2\+4y^2-2}}\ⅆy\ⅆx$

$\mathrm{evalf}\left(q\right)$ = $2.155517778$$$