The SurfaceInt(f, domain) calling sequence computes the surface integral of the function $f$ over the surface specified on the right-hand side of domain.  The left-hand side of domain is the list of variables of the function $f$.

If integral is specified for output_type, the command returns an unevaluated integral; if value is specified, the command returns the computed value of the integral. The default value for output_type is value.

Specify the surface domain using unevaluated function calls. The possible surfaces are Box, Sphere, and Surface.

$\mathrm{Box}\left(r_1\,r_2\,r_3\right)$

Each $r_i$ must have type algebraic..algebraic.  These represent the sides of the box. The surface integral is taken over each face of the box.

Sphere(center, radius)

The first parameter of Sphere, center, must have type 'Vector'(3, algebraic). The second parameter radius must have type algebraic. These represent the center and radius of the sphere, respectively.  If a coordinate system attribute is specified on center, the center is interpreted in this coordinate system.

Surface(v, range, coordinate_system)

The first argument, $v$, must have type 'Vector'(3, algebraic). The second argument, range, can be:

* [name1, name2] = region(arguments) where region is any two-dimensional region that Student[VectorCalculus][int] accepts: Circle, Ellipse, Rectangle, Region, Sector, or Triangle.

* name1=range1, name2=range2  This explicitly specifies the ranges for the two parameters.

If the optional final argument, coordinate_system, is specified, it must be an equation of the form coords=name or coordinates=name, where name is a recognized coordinate system. $v$ is interpreted in the specified coordinate system.

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

$\mathrm{SurfaceInt}\left(1\&comma;\left[x\&comma;y\&comma;z\right]=\mathrm{Surface}\left(⟨r\&comma;s\&comma;t⟩\&comma;s=0..\mathrm{\pi }\&comma;t=0..2\mathrm{\pi }\&comma;\mathrm{coords}=\mathrm{spherical}\right)\right)\mathbf{assuming}0<r$

$4\mathrm{\pi }{r}^2$

$\mathrm{SurfaceInt}\left(x+y+z\&comma;\left[x\&comma;y\&comma;z\right]=\mathrm{Surface}\left(⟨s\&comma;t\&comma;4-2s-t⟩\&comma;\left[s\&comma;t\right]=\mathrm{Triangle}\left(⟨0\&comma;0⟩\&comma;⟨1\&comma;0⟩\&comma;⟨0\&comma;1⟩\right)\right)\right)$

$\frac{11\sqrt{6}}{6}$

$\mathrm{SurfaceInt}\left(y^2\&comma;\left[x\&comma;y\&comma;z\right]=\mathrm{Sphere}\left(⟨0\&comma;0\&comma;0⟩\&comma;r\right)\right)$

$\frac{4\mathrm{\pi }{r}^4}{3}$

$\mathrm{SurfaceInt}\left(xyz\&comma;\left[x\&comma;y\&comma;z\right]=\mathrm{Box}\left(1..2\&comma;3..4\&comma;5..6\right)\right)$

$\frac{693}{4}$

$\mathrm{SurfaceInt}\left(xyz\&comma;\left[x\&comma;y\&comma;z\right]=\mathrm{Box}\left(1..2\&comma;3..4\&comma;5..6\right)\&comma;'\mathrm{output}'='\mathrm{integral}'\right)$

${\int }_5^6\left({\int }_3^{4}3st\&DifferentialD;s\right)\&DifferentialD;t+{\int }_5^6\left({\int }_1^{2}7st\&DifferentialD;s\right)\&DifferentialD;t+{\int }_3^4\left({\int }_1^{2}11st\&DifferentialD;s\right)\&DifferentialD;t$