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Student[MultivariateCalculus]

 CenterOfMass
 return the center of mass defined by the region and mass density function

 Calling Sequence CenterOfMass(f(x,y), x=a..b, y=c..d, opts) CenterOfMass(f(x,y,z), x=a..b, y=c..d, z=e..f, opts)

Parameters

 f(x, y), f(x, y, z) - algebraic expressions x, y, z - name; specify the independent variables a, b, c, d, e, f - algebraic; limits of integration opts - (optional) equation(s) of the form option=value where option is one of coordinates, functionoptions, output, pointoptions, showfunction or showregion; specify output options

Description

 • The CenterOfMass(f(x,y),x=a..b,y=c..d) calling sequence returns the center of mass of a two-dimensional region defined by the limits of integration and the mass density function $f\left(x,y\right)$. All values or plots are returned in the coordinates specified, which is cartesian coordinates by default. For the three-dimensional case, use the CenterOfMass(f(x,y,z),x=a..b,y=c..d,z=e..f) calling sequence.
 • The opts argument can contain any of the following equations that set output options.
 coordinates = cartesian[x,y], polar[r,theta] (2-D), cartesian[x,y,z], cylindrical[r,theta,z], spherical[r,phi,theta]
 Determines the coordinate system being used. The first variable of polar, cylindrical, and spherical is assumed to be the radial component. The default is cartesian.
 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 center of mass is returned. Plot options are ignored if output = value. The default is output = value.
 * output = plot specifies that a plot displays, which shows the region, the density function, and the center of mass.
 * output = integral specifies that the inert form of the center of mass integral displays.
 pointoptions = list
 Specifies the plot options for plotting the point of the center of mass. For more information on plotting options, see plot3d/options.
 showfunction = true or false
 Determines whether the density function is plotted. The default is true.
 showregion = true or false
 Determines whether the region is plotted. The default 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.
 • For information on how to change the default colors, see the Student[SetColors] help page.
 • Note: The order of integration is determined by the order of the input ranges if the limits of integration are constant. If the region over which the integration takes place has curved boundaries, the CenterOfMass command will modify the order of integration to form a correct integral based on the given input ranges.

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{MultivariateCalculus}]\right):$
 > $\mathrm{CenterOfMass}\left({x}^{2}+{y}^{2},x=1..4,y=-1..6\right)$
 $\frac{{565}}{{208}}{,}\frac{{765}}{{208}}$ (1)
 > $\mathrm{CenterOfMass}\left(x+z,x=2..4,y=x-1..x+6,z=-2..4+y,\mathrm{output}=\mathrm{integral}\right)$
 $\frac{{{∫}}_{{2}}^{{4}}{{∫}}_{{x}{-}{1}}^{{x}{+}{6}}{{∫}}_{{-}{2}}^{{4}{+}{y}}{x}{}\left({x}{+}{z}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}{{{∫}}_{{2}}^{{4}}{{∫}}_{{x}{-}{1}}^{{x}{+}{6}}{{∫}}_{{-}{2}}^{{4}{+}{y}}\left({x}{+}{z}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}{,}\frac{{{∫}}_{{2}}^{{4}}{{∫}}_{{x}{-}{1}}^{{x}{+}{6}}{{∫}}_{{-}{2}}^{{4}{+}{y}}{y}{}\left({x}{+}{z}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}{{{∫}}_{{2}}^{{4}}{{∫}}_{{x}{-}{1}}^{{x}{+}{6}}{{∫}}_{{-}{2}}^{{4}{+}{y}}\left({x}{+}{z}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}{,}\frac{{{∫}}_{{2}}^{{4}}{{∫}}_{{x}{-}{1}}^{{x}{+}{6}}{{∫}}_{{-}{2}}^{{4}{+}{y}}{z}{}\left({x}{+}{z}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}{{{∫}}_{{2}}^{{4}}{{∫}}_{{x}{-}{1}}^{{x}{+}{6}}{{∫}}_{{-}{2}}^{{4}{+}{y}}\left({x}{+}{z}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}$ (2)
 > $\mathrm{CenterOfMass}\left(r,t=0..\frac{\mathrm{π}}{2},r=0..5,\mathrm{coordinates}={\mathrm{polar}}_{r,t},\mathrm{output}=\mathrm{plot},\mathrm{pointoptions}=\left[\mathrm{color}=\mathrm{green}\right],\mathrm{orientation}=\left[50,70,-5\right]\right)$