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Average Value of a Function: $f\left(x,y\right)$

 Description In the Cartesian coordinate system, determine the average value of a bivariate function.

Average Value of a Function: $f\left(x,y\right)$

Function

 > ${{x}}^{{3}}{y}$
 ${{x}}^{{3}}{}{y}$ (1)

Region: $\left\{u\left(x\right)\le y\le v\left(x\right),a\le x\le b\right\}$

$u\left(x\right)$

 > ${{x}}^{{2}}$
 ${{x}}^{{2}}$ (2)

$v\left(x\right)$

 > ${x}$
 ${x}$ (3)

$a$

 > ${0}$
 ${0}$ (4)

$b$

 > ${1}$
 ${1}$ (5)

Inert integral:

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{FunctionAverage}\right]\left(,{y}=..,{x}=..,\mathrm{output}=\mathrm{integral}\right)$
 ${6}{}\left({{∫}}_{{0}}^{{1}}{{∫}}_{{{x}}^{{2}}}^{{x}}{{x}}^{{3}}{}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)$ (6)

Value

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{FunctionAverage}\right]\left(,{y}=..,{x}=..\right)$
 $\frac{{1}}{{8}}$ (7)

 Commands Used