The SurfaceArea(f(x,y), x=a..b, y=c..d) calling sequence returns the value of the surface area defined by z=f(x,y) over the specified region, if this can be exactly determined, or the integral representing this value. In the latter case, use the evalf function to obtain a numerical approximation to the exact result.

The output parameter can be used to select whether this command returns a value, the integral representing the surface area, or a plot showing the surface and region over which the area is to be considered.

If the output=plot option is specified, the function f(x,y) is plotted over a region slightly larger than that specified by the second and third parameters.  The part of the surface over that specified region is colored differently from the part outside that region. Vertical lines are drawn at the corner of the region.

The opts and sopts arguments can contain any or some of the following equations that set output options. The valid options are described in the parameters section.

areaoptions = list Specifies the plot options for plotting the portion of the surface that lies over the selected region. For more information on plotting options, see plot3d/options.

edgeoptions = list Specifies the plot options for plotting the lines marking the corners of the region over which the surface area is to be computed. For more information on plotting options, see plot3d/options.

functionoptions = list Specifies the plot options for plotting the function $f\left(x\,y\right)$. For more information on plotting options, see plot3d/options.

output = value, plot, or integral This option controls the return value of the function.

output = value specifies that the value of the surface area is returned. Plot options are ignored if output = value.  The default is output = value.

output = plot specifies that a plot is displayed showing the a graph of the expression, the specified region, and the portion of the surface over that region.

output = integral specifies that the inert form of the surface area integral is returned.  For this selection only, the endpoints of the integration ranges can be arbitrary algebraic expressions.

regionoptions = list Specifies the plot options for the region over which the area of the surface is being considered. For more information on plotting options, see plot3d/options.

showarea = true or false Specifies whether the area of the surface over the selected region is plotted (distinctly from the surface). The default is true.

showedges = true or false Determines whether the lines marking the region corners are plotted. The default is true.

showfunction = true or false Determines whether the function is plotted.  When true, the function is plotted over a region slightly larger than the region over which the area is being considered. The default is true.

showregion = true or false Determines whether the region is plotted. The default is true.

The SurfaceArea(surf) calling sequence returns the value of the surface area defined by surf, if this can be exactly determined, or the integral representing this value.

Specify the surface surf 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[MultivariateCalculus][MultiInt] accepts: Circle, Ellipse, Rectangle, Region, Sector, or Triangle.

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

caption = anything

A caption for the plot.

The default caption is constructed from the parameters and the command options. caption = "" disables the default caption. For more information about specifying a caption, see plot/typesetting.

title = anything

A title for the plot.

The default title is constructed from the parameters and the command options. title = "" disables the default title. For more information about specifying a title, see plot/typesetting.

For information on how to change the default colors, see the Student[SetColors] help page.

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

$\mathrm{SurfaceArea}\left(x^2+y\,x=a..b\,y=c..d\,\mathrm{output}=\mathrm{integral}\right)$

${\int }_a^b\left({\int }_c^d\sqrt{4x^2+2}\ⅆy\right)\ⅆx$

$\mathrm{SurfaceArea}\left(x^2+y\,x=0..1\,y=0..1\right)$

$\frac{\sqrt{6}}{2}+\frac{\mathrm{arcsinh}\left(\sqrt{2}\right)}{2}$

$\mathrm{SurfaceArea}\left(x^2+y\,x=0..1\,y=0..1\,\mathrm{output}=\mathrm{plot}\,\mathrm{functionoptions}=\left[\mathrm{transparency}=0.8\right]\right)$

$\mathrm{SurfaceArea}\left(x^2+y\,x=0..1\,y=0..1\,\mathrm{output}=\mathrm{plot}\right)$

$\mathrm{SurfaceArea}\left(\mathrm{Box}\left(1..2\,2..3\,3..4\right)\right)$

$6$

$\mathrm{SurfaceArea}\left(\mathrm{Box}\left(1..2\,3..5\,6..9\right)\,\mathrm{output}=\mathrm{plot}\right)$

$\mathrm{SurfaceArea}\left(\mathrm{Sphere}\left(⟨a\,b\,c⟩\,r\right)\right)$

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

$\mathrm{SurfaceArea}\left(\mathrm{Sphere}\left(⟨0\,0\,0⟩\,3\right)\,\mathrm{output}=\mathrm{plot}\right)$

$\mathrm{SurfaceArea}\left(\mathrm{Surface}\left(⟨s\,t\,1⟩\,s=0..1\,t=0..s\right)\right)$

$\frac{1}{2}$

$\mathrm{SurfaceArea}\left(\mathrm{Surface}\left(⟨s\,t\,1⟩\,\left[s\,t\right]=\mathrm{Triangle}\left(⟨0\,0⟩\,⟨1\,0⟩\,⟨0\,1⟩\right)\right)\right)$

$\frac{1}{2}$

$\mathrm{SurfaceArea}\left(\mathrm{Surface}\left(⟨s\,t\,1+s^2{t}^2⟩\,\left[s\,t\right]=\mathrm{Sector}\left(\mathrm{Ellipse}\left(⟨0\,0⟩\,2\,1\,0\right)\,-\frac{\mathrm{\pi }}{4}\,\frac{\mathrm{\pi }}{4}\right)\right)\,\mathrm{output}=\mathrm{plot}\right)$