$f\=x y \left(x^2\+y^2\right) \sqrt{1\+4 x^2\+4 y^2}$

so that the surface integral itself is given from first principles in Cartesian coordinates by

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$⟨r \cos \left(\mathrm{\θ}\right)\,r \sin \left(\mathrm{\θ}\right)\,r^2⟩$$\stackrel{\text{assign to a name}}{\to }$$R$

Obtain $∥{\mathbf{R}}_r\times {\mathbf{R}}_{\mathrm{\θ}}∥$ 

$∥\left(\frac{\partial }{\partial r} \mathbf{R}\right)\times \left(\frac{\partial }{\partial \mathrm{\θ}} \mathbf{R}\right)∥$ = $\sqrt{4r^4{\cos \left(\mathrm{\theta }\right)}^2+4r^4{\sin \left(\mathrm{\theta }\right)}^2+{\left({\cos \left(\mathrm{\theta }\right)}^{2}r+{\sin \left(\mathrm{\theta }\right)}^{2}r\right)}^2}$$\stackrel{\text{assuming positive}}{\to }$$r\sqrt{4r^2+1}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{dsig}$

Obtain the integrand in polar coordinates

$\left(r \cos \left(\mathrm{\θ}\right)\right) \left(r \sin \left(\mathrm{\θ}\right)\right) \left(r^2\right) \mathrm{dsig}$ = $r^5\cos \left(\mathrm{\theta }\right)\sin \left(\mathrm{\theta }\right)\sqrt{4r^2+1}$$\stackrel{\text{assign to a name}}{\to }$$G$

Form and evaluate the surface integral

${\int }_0^{\mathrm{\π}\/4}{\int }_0^{1\/\sqrt{{\cos }^2\left(\mathrm{\θ}\right)\+4 {\sin }^2\left(\mathrm{\θ}\right)}}G \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{5}{504}\sqrt{5}-\frac{1}{3360}-\frac{169}{1260000}\sqrt{5}\sqrt{13}$$\stackrel{\text{at 10 digits}}{\to }$$0.02080422869$

Table 9.6.9(a)   Solution from first principles implemented in polar coordinates

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$\mathbf{R}≔⟨r \cos \left(\mathrm{\θ}\right)\,r \sin \left(\mathrm{\θ}\right)\,r^2⟩\:$

Use the CrossProduct, Norm, and simplify commands to obtain $∥{\mathbf{R}}_r\times {\mathbf{R}}_{\mathrm{\θ}}∥$ 

$\mathrm{dsig}≔\mathrm{simplify}\left(\mathrm{Norm}\left(\mathrm{CrossProduct}\left(\mathrm{diff}\left(\mathbf{R}\,r\right)\,\mathrm{diff}\left(\mathbf{R}\,\mathrm{\θ}\right)\right)\right)\right) assuming r\>0$

$r\sqrt{4r^2\+1}$

Form and evaluate the surface integral

$G≔\left(r \cos \left(\mathrm{\theta }\right)\right) \left(r \sin \left(\mathrm{\theta }\right)\right) \left(r^2\right) \mathrm{dsig}$

$r^5\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\θ}\right)\sqrt{4r^2\+1}$

$q≔\mathrm{Int}\left(G\,\left[r\=0..1\/\sqrt{{\cos }^2\left(\mathrm{\θ}\right)\+4 {\sin }^2\left(\mathrm{\θ}\right)}\,\mathrm{\θ}\=0..\mathrm{\π}\/4\right]\right)$

${\∫}_0^{\frac{1}{4}\mathrm{\π}}{\∫}_0^{\frac{1}{\sqrt{{\cos \left(\mathrm{\θ}\right)}^2\+4{\sin \left(\mathrm{\θ}\right)}^2}}}{r}^5\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\θ}\right)\sqrt{4r^2\+1}\ⅆr\ⅆ\mathrm{\θ}$

$\mathrm{value}\left(q\right)$

$\frac{5}{504}\sqrt{5}-\frac{1}{3360}-\frac{169}{1260000}\sqrt{5}\sqrt{13}$

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

$0.02080422869$

Table 9.6.9(b)   Solution from first principles implemented in polar coordinates

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$E≔x^2\+4 y^2\=1\:$

$Y≔\mathrm{solve}\left(E\,y\right)$

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

$\mathrm{solve}\left(\left\{E\,y\=x\right\}\,\mathrm{explicit}\right)\left[1\right]$

$\left\{x\=\frac{1}{5}\sqrt{5}\,y\=\frac{1}{5}\sqrt{5}\right\}$

$G≔x y \left(x^2\+y^2\right) \sqrt{1\+4 x^2\+4 y^2}\:$

Form and evaluate the surface integral (See Figure 9.6.9(a))

$q≔\mathrm{Int}\left(G\,\left[y\=0..x\,x\=0..1\/\sqrt{5}\right]\right)\+\mathrm{Int}\left(G\,\left[y\=0..Y_1\,x\=1\/\sqrt{5}..1\right]\right)$

${\∫}_0^{\frac{1}{5}\sqrt{5}}{\∫}_0^{x}xy\left(x^2\+y^2\right)\sqrt{4x^2\+4y^2\+1}\ⅆy\ⅆx\+{\∫}_{\frac{1}{5}\sqrt{5}}^1{\∫}_0^{\frac{1}{2}\sqrt{-x^2\+1}}xy\left(x^2\+y^2\right)\sqrt{4x^2\+4y^2\+1}\ⅆy\ⅆx$

$\mathrm{value}\left(q\right)$

$-\frac{1}{3360}-\frac{169}{1260000}\sqrt{5}\sqrt{13}\+\frac{5}{504}\sqrt{5}$

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

$0.02080422869$

Table 9.6.9(c)   Solution from first principles implemented in Cartesian coordinates