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Iterated Triple Integral in Cartesian Coordinates

 Description Calculate the iterated triple integral in Cartesian coordinates.



Iterated Triple Integrals in Cartesian Coordinates

Integrand:

 >
 ${x}{}{y}{}{z}$ (1)

Region: $\left\{{z}_{1}\left(x,y\right)\le z\le {z}_{2}\left(x,y\right),{y}_{1}\left(x\right)\le y\le {y}_{2}\left(x\right),a\le x\le b\right\}$

${z}_{1}\left(x,y\right)$

 > ${1}{-}{x}{-}{y}$
 ${1}{-}{x}{-}{y}$ (2)

${z}_{2}\left(x,y\right)$

 > ${10}{-}{{x}}^{{2}}{-}{{y}}^{{2}}$
 ${10}{-}{{x}}^{{2}}{-}{{y}}^{{2}}$ (3)

${y}_{1}\left(x\right)$

 > ${{x}}^{{2}}$
 ${{x}}^{{2}}$ (4)

${y}_{2}\left(x\right)$

 > ${x}$
 ${x}$ (5)

$a$

 > ${0}$
 ${0}$ (6)

$b$

 > ${1}$
 ${1}$ (7)

Inert Integral:

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{MultiInt}\right]\left(,{z}=..,{y}=..,{x}=..,\mathrm{output}=\mathrm{integral}\right)$
 ${{∫}}_{{0}}^{{1}}{{∫}}_{{{x}}^{{2}}}^{{x}}{{∫}}_{{1}{-}{x}{-}{y}}^{{10}{-}{{x}}^{{2}}{-}{{y}}^{{2}}}{x}{}{y}{}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (8)

Value:

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{MultiInt}\right]\left(,{z}=..,{y}=..,{x}=..\right)$
 $\frac{{26081}}{{15120}}$ (9)

Stepwise Evaluation:

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{MultiInt}\right]\left(,{z}=..,{y}=..,{x}=..,\mathrm{output}=\mathrm{steps}\right)$
 $\begin{array}{ccc}\multicolumn{3}{c}{{{\int }}_{{0}}^{{1}}{{\int }}_{{{x}}^{{2}}}^{{x}}{{\int }}_{{1}{-}{x}{-}{y}}^{{10}{-}{{x}}^{{2}}{-}{{y}}^{{2}}}{x}{}{y}{}{z}\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}}\\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {{\int }}_{{0}}^{{1}}{{\int }}_{{{x}}^{{2}}}^{{x}}\left(\genfrac{}{}{0}{}{\frac{{x}{}{y}{}{{z}}^{{2}}}{{2}}}{\phantom{{z}{=}{1}{-}{x}{-}{y}{..}{10}{-}{{x}}^{{2}}{-}{{y}}^{{2}}}}{|}\genfrac{}{}{0}{}{\phantom{\frac{{x}{}{y}{}{{z}}^{{2}}}{{2}}}}{{z}{=}{1}{-}{x}{-}{y}{..}{10}{-}{{x}}^{{2}}{-}{{y}}^{{2}}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\hfill \\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {{\int }}_{{0}}^{{1}}{{\int }}_{{{x}}^{{2}}}^{{x}}\frac{{x}{}{y}{}\left({\left({10}{-}{{x}}^{{2}}{-}{{y}}^{{2}}\right)}^{{2}}{-}{\left({1}{-}{x}{-}{y}\right)}^{{2}}\right)}{{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\hfill \\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {{\int }}_{{0}}^{{1}}\left(\genfrac{}{}{0}{}{\frac{{x}{}\left(\frac{{{y}}^{{6}}}{{6}}{+}\frac{\left({-}{21}{+}{2}{}{{x}}^{{2}}\right){}{{y}}^{{4}}}{{4}}{+}\frac{\left({2}{-}{2}{}{x}\right){}{{y}}^{{3}}}{{3}}{+}\frac{\left({\left({10}{-}{{x}}^{{2}}\right)}^{{2}}{-}{\left({1}{-}{x}\right)}^{{2}}\right){}{{y}}^{{2}}}{{2}}\right)}{{2}}}{\phantom{{y}{=}{{x}}^{{2}}{..}{x}}}{|}\genfrac{}{}{0}{}{\phantom{\frac{{x}{}\left(\frac{{{y}}^{{6}}}{{6}}{+}\frac{\left({-}{21}{+}{2}{}{{x}}^{{2}}\right){}{{y}}^{{4}}}{{4}}{+}\frac{\left({2}{-}{2}{}{x}\right){}{{y}}^{{3}}}{{3}}{+}\frac{\left({\left({10}{-}{{x}}^{{2}}\right)}^{{2}}{-}{\left({1}{-}{x}\right)}^{{2}}\right){}{{y}}^{{2}}}{{2}}\right)}{{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}}\left(\frac{{x}{}\left({{x}}^{{6}}{-}{{x}}^{{12}}\right)}{{12}}{+}\frac{{x}{}\left({-}{21}{+}{2}{}{{x}}^{{2}}\right){}\left({{x}}^{{4}}{-}{{x}}^{{8}}\right)}{{8}}{+}\frac{{x}{}\left({2}{-}{2}{}{x}\right){}\left({{x}}^{{3}}{-}{{x}}^{{6}}\right)}{{6}}{+}\frac{{x}{}\left({\left({10}{-}{{x}}^{{2}}\right)}^{{2}}{-}{\left({1}{-}{x}\right)}^{{2}}\right){}\left({{x}}^{{2}}{-}{{x}}^{{4}}\right)}{{4}}\right)\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}}{{168}}{}{{x}}^{{14}}{+}\frac{{11}}{{16}}{}{{x}}^{{8}}{-}\frac{{1}}{{48}}{}{{x}}^{{12}}{+}\frac{{19}}{{80}}{}{{x}}^{{10}}{-}\frac{{791}}{{144}}{}{{x}}^{{6}}{+}\frac{{1}}{{27}}{}{{x}}^{{9}}{+}\frac{{1}}{{6}}{}{{x}}^{{5}}{-}\frac{{1}}{{14}}{}{{x}}^{{7}}{+}\frac{{99}}{{16}}{}{{x}}^{{4}}\right)}{\phantom{{x}{=}{0}{..}{1}}}{|}\genfrac{}{}{0}{}{\phantom{\left({-}\frac{{1}}{{168}}{}{{x}}^{{14}}{+}\frac{{11}}{{16}}{}{{x}}^{{8}}{-}\frac{{1}}{{48}}{}{{x}}^{{12}}{+}\frac{{19}}{{80}}{}{{x}}^{{10}}{-}\frac{{791}}{{144}}{}{{x}}^{{6}}{+}\frac{{1}}{{27}}{}{{x}}^{{9}}{+}\frac{{1}}{{6}}{}{{x}}^{{5}}{-}\frac{{1}}{{14}}{}{{x}}^{{7}}{+}\frac{{99}}{{16}}{}{{x}}^{{4}}\right)}}{{x}{=}{0}{..}{1}}\hfill \end{array}$
 $\frac{{26081}}{{15120}}$ (10)

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