Volume of Revolution - Maple Help

Student[Calculus1]

 VolumeOfRevolution
 find the volume of revolution of a curve

 Calling Sequence VolumeOfRevolution(f(x), x = a..b, opts) VolumeOfRevolution(f(x), g(x), x = a..b, opts) VolumeOfRevolution(f(x), a..b, opts) VolumeOfRevolution(f(x), g(x), a..b, opts)

Parameters

 f(x), g(x) - algebraic expressions in variable x x - name; specify the independent variable a,b - algebraic expressions; specify the endpoints of the curve opts - equation(s) of the form option=value where option is one of axis, distancefromaxis, functionoptions, function2options, lineoptions, method, numpoints, output, partition, regionoptions, revolutionpoints, showfunction, showrotationline, showsum, showvolume, sumvolumeoptions, volumeoptions, volume2options, border, or Student plot options; specify output options

Description

 • The VolumeOfRevolution(f(x), x=a..b) command returns the volume of revolution of the expression f(x) from a to b.
 • The VolumeOfRevolution(f(x), g(x), x=a..b) command returns the volume of revolution between the expressions f(x) and g(x) from a to b.
 • By using the output option, you can specify that the command returns a plot or inert integral instead.
 • 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, method, 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.
 – function2options = list A list of options for the plot of the expression g(x). By default, the expression is plotted as a solid magenta 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 turquoise line. For more information on plot options, see plot3d/option.
 – method = left, midpoint, right, upper, lower The method option specifies the form the approximating Riemann sum should take.
 – numpoints = posint The number of points plotted along the horizontal (or vertical) axis. By default, this value is 50.
 – output = value, integral, region, sum, sumvalue, plot, or animation This option controls the return value of the function.
 • output = value specifies that the volume is returned. 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 Riemann sum of the approximation is returned. Plot options are ignored if output = sum.
 • output = sumvalue specifies that the value of the approximation is returned. Plot options are ignored if output = sumvalue
 • output = plot specifies that a plot showing the expression or expressions and their rotations around the horizontal (or vertical) axis is displayed.
 • output = animation specifies that an animation, which steps through the separate disks or cylindrical shells of the approximating sum, is returned.
 • output = region specifies that a plot showing the slice that is about to be rotated 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 option is ignored.
 – regionoptions = list A list of options for the slice that is rotated.  For more information on plot options, see plot3d/option.
 – revolutionpoints = posint The number of points plotted around the volume (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 Riemann sum is plotted. Depending on axis, the sum is displayed by the use of disks (axis=horizontal) or cylindrical shells (axis=vertical). By default, the value is false.  When output=animation, this option is ignored.
 – showvolume = true or false Whether the volume is plotted. By default, the value is true.
 – sumvolumeoptions = list A list of options for the approximating Riemann sum. For more information on plot options, see plot3d/option.
 – volumeoptions = list A list of options for the volume of revolution generated by the expression f(x). For more information on plot options, see plot3d/option.
 – volume2options = list A list of options for the volume of revolution generated by the expression g(x). For more information on plot options, see plot3d/option.
 – border = true or false Adds a black border to the cylindrical shells when the following options are set: axis=vertical, showsum=true, output=plot. By default, the value is true.
 – 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.

Notes

 • If axis = vertical and only one function is supplied, the resulting volume of revolution will be bounded by the function, the specified range, and the horizontal axis (y=0).

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Calculus1}\right]\right):$
 > $\mathrm{VolumeOfRevolution}\left({x}^{2}+1,x=0..1\right)$
 $\frac{{28}{}{\mathrm{\pi }}}{{15}}$ (1)
 > $\mathrm{VolumeOfRevolution}\left({x}^{10}+1,{x}^{2}+1,x=0..1,\mathrm{distancefromaxis}=-1\right)$
 $\frac{{432}{}{\mathrm{\pi }}}{{385}}$ (2)
 > $\mathrm{VolumeOfRevolution}\left(\mathrm{sin}\left(x\right)+1,x=0..3,\mathrm{output}=\mathrm{integral}\right)$
 ${{\int }}_{{0}}^{{3}}{\mathrm{\pi }}{}{\left({\mathrm{sin}}{}\left({x}\right){+}{1}\right)}^{{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (3)
 > $\mathrm{Int}\left(\mathrm{\pi }{\left(\mathrm{sin}\left(x\right)+1\right)}^{2},x=0..3\right)$
 ${{\int }}_{{0}}^{{3}}{\mathrm{\pi }}{}{\left({\mathrm{sin}}{}\left({x}\right){+}{1}\right)}^{{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (4)
 > $\mathrm{VolumeOfRevolution}\left(\mathrm{sin}\left(x\right)+1,x=2\mathrm{\pi }..3\mathrm{\pi },\mathrm{output}=\mathrm{integral},\mathrm{axis}=\mathrm{vertical}\right)$
 ${{\int }}_{{2}{}{\mathrm{\pi }}}^{{3}{}{\mathrm{\pi }}}{2}{}{\mathrm{\pi }}{}{x}{}\left({\mathrm{sin}}{}\left({x}\right){+}{1}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (5)
 > $\mathrm{VolumeOfRevolution}\left(\mathrm{cos}\left(x\right)+1,x=0..4\mathrm{\pi },\mathrm{output}=\mathrm{plot}\right)$
 > $\mathrm{VolumeOfRevolution}\left(\mathrm{cos}\left(x\right)+3,\mathrm{sin}\left(x\right)+2,x=0..4\mathrm{\pi },\mathrm{output}=\mathrm{plot}\right)$
 > $\mathrm{VolumeOfRevolution}\left(\mathrm{sin}\left(x\right)+1,x=2\mathrm{\pi }..3\mathrm{\pi },\mathrm{output}=\mathrm{plot},\mathrm{axis}=\mathrm{vertical},\mathrm{distancefromaxis}=\mathrm{\pi },\mathrm{method}=\mathrm{midpoint},\mathrm{partition}=3,\mathrm{showsum}\right)$
 > $\mathrm{VolumeOfRevolution}\left({x}^{2},x=-1..1,\mathrm{output}=\mathrm{plot},\mathrm{showregion}=\mathrm{true}\right)$
 > 

Compatibility

 • The regionoptions option was introduced in Maple 16.
 • The output option was updated in Maple 16.