The SurfaceOfRevolution(f(x), x=a..b) command returns the lateral surface area of the surface of revolution generated by rotating the expression f(x) from a to b around the horizontal or vertical axis, or a line parallel to one of the axes. By using the output option, you can specify that the command returns a plot or inert integral instead.

In general, this lateral surface area cannot be represented in closed form. Also, most closed forms contain special functions that are beyond the normal scope of a first course in single-variable calculus.

If the independent variable can be uniquely determined from the expression, the parameter x need not be included in the calling sequence.

The opts argument can contain any of the Student plot options or any of the following options. The options axis, distancefromaxis, output, and partition specify the volume of revolution that is computed.  The remaining options set plot options.

axis = horizontal or vertical Whether the expression is rotated horizontally or vertically. By default, the rotation is horizontal.

distancefromaxis = numeric The distance of the line of rotation from the coordinate axis. For example, distancefromaxis=-2 rotates around the line y=-2 (or x=-2). By default, the value is 0. That is, the rotation is around the coordinate axis.

functionoptions = list A list of options for the plot of the expression f(x). By default, the expression is plotted as a solid blue line. For more information on plot options, see plot3d/option.

lineoptions = list A list of options for the plot of the line of rotation. By default, the expression is plotted as a solid green line. For more information on plot options, see plot3d/option.

numpoints = posint The number of points plotted along the horizontal (or vertical) axis. By default, this value is 50.

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

output = value specifies that the lateral area of the surface of revolution is returned. (That is, the area traced by the rotated graph of f(x); the area of the end "caps" is not computed.) Plot options are ignored if output = value. This is the default.

output = integral specifies that an inert integral with the appropriate integrand is returned. Plot options are ignored if output = integral.

output = sum specifies that an inert sum of the approximating frustums of cones is returned. Plot options are ignored if output = sum.

output = plot specifies that a plot showing the expression and its rotation around the horizontal (or vertical) axis, is displayed.

partition = posint The partition option specifies the number of equally spaced subintervals in (a, b), if the output contains any form of approximating sum. If not, this parameter is ignored.

revolutionpoints = posint The number of points plotted around the surface (that is, along the circumferences of the circles traced by the rotating curve). By default, this value is 40.

showfunction = true or false Whether the expression f(x) (or both expressions f(x) and g(x)) is plotted. By default, the value is true.

showrotationline = true or false Whether the line of rotation is plotted. By default, the value is true.

showsum = true or false Whether the approximating sum is plotted. By default, the value is false.

showsurface = true or false Whether the surface is plotted. By default, the value is true.

sumsurfaceoptions = list A list of options for the approximating sum. For more information on plot options, see plot3d/option.

surfaceoptions = list A list of options for the surface of revolution generated by the expression f(x). For more information on plot options, see plot3d/option.

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.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Calculus1}\right]\right)\:$

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

$\frac{15\mathrm{\pi }\ln \left(2\right)}{16}+\frac{25\mathrm{\pi }\sqrt{5}}{16}+\frac{15\mathrm{\pi }\ln \left(\frac{1}{2}+\frac{\sqrt{5}}{4}\right)}{32}$

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

$\frac{17\mathrm{\pi }\ln \left(2\right)}{16}+\frac{7\mathrm{\pi }\sqrt{5}}{16}+\frac{17\mathrm{\pi }\ln \left(\frac{1}{2}+\frac{\sqrt{5}}{4}\right)}{32}$

$\mathrm{SurfaceOfRevolution}\left(\sin \left(x\right)+1\,x=0..3\,\mathrm{output}=\mathrm{integral}\right)$

${\int }_0^{3}2\mathrm{\pi }\left(\sin \left(x\right)+1\right)\sqrt{1+{\cos \left(x\right)}^2}\ⅆx$

$\mathrm{SurfaceOfRevolution}\left(x^2\,x=1..2\,\mathrm{axis}=\mathrm{vertical}\right)$

$-\frac{5\mathrm{\pi }\sqrt{5}}{6}+\frac{17\sqrt{17}\mathrm{\pi }}{6}$

$\mathrm{SurfaceOfRevolution}\left(\cos \left(x\right)+1\,x=0..4\mathrm{\pi }\,\mathrm{output}=\mathrm{plot}\right)$

$\mathrm{SurfaceOfRevolution}\left(1+x\left(3-x\right)\,x=0..3\,\mathrm{output}=\mathrm{plot}\,\mathrm{surfaceoptions}=\left[\mathrm{shading}=Z\right]\,\mathrm{functionoptions}=\left[\mathrm{color}=\mathrm{white}\right]\right)$

$\mathrm{SurfaceOfRevolution}\left(\frac{1}{x}\cos \left(x\right)\,\mathrm{\pi }..4\mathrm{\pi }\,\mathrm{output}=\mathrm{plot}\,\mathrm{axis}=\mathrm{vertical}\right)$

$\mathrm{SurfaceOfRevolution}\left(\frac{1}{x}\cos \left(x\right)\,\mathrm{\pi }..4\mathrm{\pi }\,\mathrm{output}=\mathrm{plot}\,\mathrm{axis}=\mathrm{vertical}\,\mathrm{distancefromaxis}=-\mathrm{\pi }\right)$