The FunctionAverage(f(x), x=a..b) command returns the average value of the expression $f\left(x\right)$ on the interval from a to b. By using options, 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 average value of an expression f(x) from x = a to x = b is the value of the expression:

The opts argument can contain any of the Student plot options or any of the following equations that (excluding output) set plot options.

averageoptions = list

A list of options for the plot of the average value of $f\left(x\right)$. By default, the average is plotted as a solid blue line. For more information on plot options, see plot/options.

functionoptions = list

A list of options for the plot of the expression $f\left(x\right)$.  By default, the expression is plotted as a solid red line. For more information on plot options, see plot/options.

output = value, integral, or plot

This option controls the return value of the function.

output = value specifies that the average value is returned. Other options are ignored if output = value.  This is the default.

output = integral specifies that the expression $\frac{1}{b-a}$ times the inert integral with the integrand f(x) is returned. Other options are ignored if output = integral.

output = plot specifies that a plot, which shows the expression and its average value from a to b, is displayed.

showaverage = true or false

Whether the average of the expression $f\left(x\right)$ over the range $a..b$ is plotted. By default, the value is true.

showfunction = true or false

Whether the expression $f\left(x\right)$ is plotted.  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.

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

$\mathrm{FunctionAverage}\left(x-3\,x=-1..1\right)$

$\mathrm{-3}$

$\mathrm{FunctionAverage}\left(\ln \left(x\right)\,x=1..4\right)$

$-1+\frac{8\ln \left(2\right)}{3}$

$\mathrm{FunctionAverage}\left(x^{\frac{3}{2}}\,1..4\,\mathrm{output}=\mathrm{integral}\right)$

$\frac{\left({\int }_1^4{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\ⅆx\right)}{3}$

$\mathrm{FunctionAverage}\left(\frac{1}{x}\,x=1..4\,\mathrm{output}=\mathrm{plot}\right)$

$\mathrm{FunctionAverage}\left(\sin \left(x\right)+x\,3..10\,\mathrm{output}=\mathrm{plot}\,\mathrm{functionoptions}=\left[\mathrm{thickness}=2\right]\,\mathrm{averageoptions}=\left[\mathrm{linestyle}=\mathrm{dash}\right]\right)$