Figure 9.6.4(a)   Plane region $R$ and the surface defined over $R$ 

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Calculus - Vector≻Integration≻Surface Integration≻Surface Defined over a 2-D Region

Table 9.6.4(a)   Solution via task template

Initialize

$f\left(x\,y\,z\right)\=x y z$$\stackrel{\text{assign as function}}{\to }$$f$

$x^2\+y^2$$\stackrel{\text{assign to a name}}{\to }$$Z$

Obtain $\mathrm{dsig}\=\sqrt{1\+z_x^2\+z_y^2}$ 

$\sqrt{1\+{\left(\frac{\partial }{\partial x} Z\right)}^2\+{\left(\frac{\partial }{\partial y} Z\right)}^2}$ = $\sqrt{4x^2\+4y^2\+1}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{dsig}$$$

Write and evaluate the appropriate surface integral

${\int }_0^{\mathrm{\π}\/4}{\int }_{\sin \left(x\right)}^{\cos \left(x\right)}f\left(x\,y\,Z\right) \mathrm{dsig} \mathit{\ⅆ}y \mathit{\ⅆ}x$ = ${\∫}_0^{\frac{1}{4}\mathrm{\π}}\left(-\frac{1}{20}{\left(4{\sin \left(x\right)}^2\+4x^2\+1\right)}^{3\/2}{x}^3-\frac{1}{20}{\left(4{\sin \left(x\right)}^2\+4x^2\+1\right)}^{3\/2}x{\sin \left(x\right)}^2\+\frac{1}{120}{\left(4{\sin \left(x\right)}^2\+4x^2\+1\right)}^{3\/2}x\+\frac{1}{20}x{\left(4{\cos \left(x\right)}^2\+4x^2\+1\right)}^{3\/2}{\cos \left(x\right)}^2\+\frac{1}{20}{\left(4{\cos \left(x\right)}^2\+4x^2\+1\right)}^{3\/2}{x}^3-\frac{1}{120}x{\left(4{\cos \left(x\right)}^2\+4x^2\+1\right)}^{3\/2}\right)\ⅆx$$\stackrel{\text{at 10 digits}}{\to }$$0.1006308329$$$

Table 9.6.4(b)   Solution from first principles

${\int }_{0.}^{\mathrm{\pi }\/4}{\int }_{\sin \left(x\right)}^{\cos \left(x\right)}f\left(x\,y\,Z\right) \mathrm{dsig} \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $0.1006308329$$$

Initialize

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

$Z≔x^2\+y^2\:$

Apply the SurfaceInt command to obtain the unevaluated surface integral

$\mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,Z⟩\,x\=0..\mathrm{\π}\/4\,y\=\sin \left(x\right)..\cos \left(x\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

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

Float one number in the SurfaceInt command to obtain a numeric result

$\mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,Z⟩\,x\=0..\mathrm{\π}\/4.\,y\=\sin \left(x\right)..\cos \left(x\right)\right)\right)$ = $0.1006308329$$$

Table 9.6.4(c)   Solution via the SurfaceInt command