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Iterated Triple Integral in Spherical Coordinates

 Description Compute the iterated triple integral in spherical coordinates.

Iterated Triple Integral in Spherical Coordinates

($\mathrm{φ}$ = colatitude, measured down from $z$-axis)

Integrand:

 > ${\mathrm{\rho }}$
 ${\mathrm{ρ}}$ (1)

Region: $\left\{{\mathrm{ρ}}_{1}\left(\mathrm{φ},\mathrm{θ}\right)\le \mathrm{ρ}\le {\mathrm{\rho }}_{2}\left(\mathrm{φ},\mathrm{θ}\right),{\mathrm{φ}}_{1}\left(\mathrm{θ}\right)\le \mathrm{φ}\le {\mathrm{φ}}_{2}\left(\mathrm{θ}\right),a\le \mathrm{θ}\le b\right\}$

${\mathrm{\rho }}_{1}\left(\mathrm{φ},\mathrm{θ}\right)$

 > ${0}$
 ${0}$ (2)

${\mathrm{\rho }}_{2}\left(\mathrm{φ},\mathrm{θ}\right)$

 > ${1}$
 ${1}$ (3)

${\mathrm{φ}}_{1}\left(\mathrm{θ}\right)$

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

${\mathrm{φ}}_{2}\left(\mathrm{θ}\right)$

 > $\frac{{\mathrm{π}}}{{6}}$
 $\frac{{1}}{{6}}{}{\mathrm{π}}$ (5)

$a$

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

$b$

 >
 ${2}{}{\mathrm{π}}$ (7)

Inert Integral:

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{MultiInt}\right]\left(,{\mathrm{ρ}}=..,{\mathrm{φ}}=..,{\mathrm{θ}}=..,\mathrm{coordinates}=\mathrm{spherical}\left[{\mathrm{ρ}}{,}{\mathrm{φ}}{,}{\mathrm{\theta }}\right],\mathrm{output}=\mathrm{integral}\right)$
 ${{∫}}_{{0}}^{{2}{}{\mathrm{π}}}{{∫}}_{{0}}^{\frac{{1}}{{6}}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{{\mathrm{ρ}}}^{{3}}{}{\mathrm{sin}}{}\left({\mathrm{φ}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{ρ}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{φ}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}$ (8)

Value:

 >
 $\frac{{1}}{{2}}{}{\mathrm{π}}{-}\frac{{1}}{{4}}{}\sqrt{{3}}{}{\mathrm{π}}$ (9)

Stepwise Evaluation:

 >
 $\begin{array}{ccc}\multicolumn{3}{c}{{{\int }}_{{0}}^{{2}{}{\mathrm{\pi }}}{{\int }}_{{0}}^{\frac{{\mathrm{\pi }}}{{6}}}{{\int }}_{{0}}^{{1}}{{\mathrm{\rho }}}^{{3}}{}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\rho }}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\phi }}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\theta }}}\\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {{\int }}_{{0}}^{{2}{}{\mathrm{\pi }}}{{\int }}_{{0}}^{\frac{{\mathrm{\pi }}}{{6}}}\left(\genfrac{}{}{0}{}{\frac{{{\mathrm{\rho }}}^{{4}}{}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right)}{{4}}}{\phantom{{\mathrm{\rho }}{=}{0}{..}{1}}}{|}\genfrac{}{}{0}{}{\phantom{\frac{{{\mathrm{\rho }}}^{{4}}{}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right)}{{4}}}}{{\mathrm{\rho }}{=}{0}{..}{1}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\phi }}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\theta }}\hfill \\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {{\int }}_{{0}}^{{2}{}{\mathrm{\pi }}}{{\int }}_{{0}}^{\frac{{\mathrm{\pi }}}{{6}}}\frac{{\mathrm{sin}}{}\left({\mathrm{\phi }}\right)}{{4}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\phi }}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\theta }}\hfill \\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {{\int }}_{{0}}^{{2}{}{\mathrm{\pi }}}\left(\genfrac{}{}{0}{}{{-}\frac{{\mathrm{cos}}{}\left({\mathrm{\phi }}\right)}{{4}}}{\phantom{{\mathrm{\phi }}{=}{0}{..}\frac{{\mathrm{\pi }}}{{6}}}}{|}\genfrac{}{}{0}{}{\phantom{{-}\frac{{\mathrm{cos}}{}\left({\mathrm{\phi }}\right)}{{4}}}}{{\mathrm{\phi }}{=}{0}{..}\frac{{\mathrm{\pi }}}{{6}}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\theta }}\hfill \\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {{\int }}_{{0}}^{{2}{}{\mathrm{\pi }}}\left(\frac{{1}}{{4}}{-}\frac{\sqrt{{3}}}{{8}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\theta }}\hfill \\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& \genfrac{}{}{0}{}{\left(\frac{{1}}{{4}}{-}\frac{\sqrt{{3}}}{{8}}\right){}{\mathrm{\theta }}}{\phantom{{\mathrm{\theta }}{=}{0}{..}{2}{}{\mathrm{\pi }}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{1}}{{4}}{-}\frac{\sqrt{{3}}}{{8}}\right){}{\mathrm{\theta }}}}{{\mathrm{\theta }}{=}{0}{..}{2}{}{\mathrm{\pi }}}\hfill \end{array}$
 $\frac{{1}}{{2}}{}{\mathrm{π}}{-}\frac{{1}}{{4}}{}\sqrt{{3}}{}{\mathrm{π}}$ (10)

 Commands Used