Cartesian 2-D - Maple Help

Iterated Double Integral in Cartesian Coordinates

 Description Compute the iterated double integral in Cartesian coordinates.

Iterated Double Integral in Cartesian Coordinates

Integrand:

 >
 ${{x}}^{{3}}{}{y}$ (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:

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

Value:

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{MultiInt}\right]\left(,{y}=..,{x}=..\right)$
 $\frac{{1}}{{48}}$ (7)

Stepwise Evaluation:

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{MultiInt}\right]\left(,{y}=..,{x}=..,\mathrm{output}=\mathrm{steps}\right)$
 $\begin{array}{ccc}\multicolumn{3}{c}{{{\int }}_{{0}}^{{1}}{{\int }}_{{{x}}^{{2}}}^{{x}}{{x}}^{{3}}{}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}\\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {{\int }}_{{0}}^{{1}}\left(\genfrac{}{}{0}{}{\frac{{{x}}^{{3}}{}{{y}}^{{2}}}{{2}}}{\phantom{{y}{=}{{x}}^{{2}}{..}{x}}}{|}\genfrac{}{}{0}{}{\phantom{\frac{{{x}}^{{3}}{}{{y}}^{{2}}}{{2}}}}{{y}{=}{{x}}^{{2}}{..}{x}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\hfill \\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {{\int }}_{{0}}^{{1}}\frac{{{x}}^{{3}}{}\left({{x}}^{{2}}{-}{{x}}^{{4}}\right)}{{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\hfill \\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& \genfrac{}{}{0}{}{\left({-}\frac{{1}}{{16}}{}{{x}}^{{8}}{+}\frac{{1}}{{12}}{}{{x}}^{{6}}\right)}{\phantom{{x}{=}{0}{..}{1}}}{|}\genfrac{}{}{0}{}{\phantom{\left({-}\frac{{1}}{{16}}{}{{x}}^{{8}}{+}\frac{{1}}{{12}}{}{{x}}^{{6}}\right)}}{{x}{=}{0}{..}{1}}\hfill \end{array}$
 $\frac{{1}}{{48}}$ (8)

 Commands Used