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Student[MultivariateCalculus]

 FunctionAverage
 return the average value of a function defined over a region

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

Parameters

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

Description

 • The FunctionAverage command returns the average value of a function over a region in one-, two-, or three-dimensional space.
 • 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], or 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.
 output = value or integral
 This option controls the return value of the function.
 * output = value specifies that the average value of the function is returned. The default is output = value.
 * output = integral specifies that the inert form of the function average is returned.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\right):$
 > $\mathrm{FunctionAverage}\left({x}^{3},x=3..4\right)$
 $\frac{{175}}{{4}}$ (1)
 > $\mathrm{FunctionAverage}\left({x}^{2}+{y}^{2},y=0..\mathrm{sqrt}\left(1-{x}^{2}\right),x=0..1\right)$
 $\frac{{1}}{{2}}$ (2)
 > $\mathrm{FunctionAverage}\left({r}^{2},r=0..1,t=0..\frac{\mathrm{\pi }}{2},\mathrm{coordinates}=\mathrm{polar}\left[r,t\right]\right)$
 $\frac{{1}}{{2}}$ (3)
 > $\mathrm{FunctionAverage}\left({x}^{2}+{y}^{2}+z,z=-2..4+{y}^{2},y=x-1..x+6,x=2..4,\mathrm{output}=\mathrm{integral}\right)$
 $\frac{{{\int }}_{{2}}^{{4}}{{\int }}_{{x}{-}{1}}^{{x}{+}{6}}{{\int }}_{{-2}}^{{{y}}^{{2}}{+}{4}}\left({{x}}^{{2}}{+}{{y}}^{{2}}{+}{z}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}{{{\int }}_{{2}}^{{4}}{{\int }}_{{x}{-}{1}}^{{x}{+}{6}}{{\int }}_{{-2}}^{{{y}}^{{2}}{+}{4}}{1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}$ (4)
 > $\mathrm{FunctionAverage}\left({x}^{2}+{y}^{2},y={x}^{2}+4..{x}^{3},x=a..b\right)$
 $\frac{{-}\frac{{64}}{{3}}{}{b}{+}\frac{{64}}{{3}}{}{a}{-}\frac{{1}}{{30}}{}{{a}}^{{10}}{+}\frac{{1}}{{30}}{}{{b}}^{{10}}{+}\frac{{1}}{{21}}{}{{a}}^{{7}}{-}\frac{{1}}{{21}}{}{{b}}^{{7}}{-}\frac{{1}}{{6}}{}{{a}}^{{6}}{+}\frac{{1}}{{6}}{}{{b}}^{{6}}{+}{{a}}^{{5}}{-}{{b}}^{{5}}{+}\frac{{20}}{{3}}{}{{a}}^{{3}}{-}\frac{{20}}{{3}}{}{{b}}^{{3}}}{{-}\frac{{1}}{{4}}{}{{a}}^{{4}}{+}\frac{{1}}{{4}}{}{{b}}^{{4}}{+}\frac{{1}}{{3}}{}{{a}}^{{3}}{-}\frac{{1}}{{3}}{}{{b}}^{{3}}{-}{4}{}{b}{+}{4}{}{a}}$ (5)
 > $\mathrm{FunctionAverage}\left({x}^{2}+{y}^{2},y={x}^{2}+4..{x}^{3},x=0..4\right)$
 $\frac{{218656}}{{175}}$ (6)
 > $\mathrm{FunctionAverage}\left({x}^{2}+{y}^{2},y={x}^{2}+4..{x}^{3},x=3..5\right)$
 $\frac{{16739158}}{{5005}}$ (7)