# Find the centroid of the region bounded by the given

Find the centroid of the region bounded by the given curves.
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Esther Phillips

$A={\int }_{0}^{27}\left(30-y\right)-{y}^{\frac{1}{3}}dy$
$A={\left(30y-\frac{{y}^{2}}{2}-\frac{3}{4}{y}^{\frac{4}{3}}\right)}_{0}^{27}$
$A=\left(30\left(27\right)-\frac{{27}^{2}}{2}-\frac{3}{4}{\left(27\right)}^{\frac{4}{3}}\right)$
$A=384.75$
${M}_{x}={\int }_{0}^{27}y\left(30-y-{y}^{\frac{1}{3}}\right)dy$
${M}_{x}={\left(15{y}^{2}-\frac{{y}^{3}}{3}-\frac{3}{7}{y}^{\frac{7}{3}}\right)}_{0}^{23}$
${M}_{x}=\frac{24057}{7}$
${M}_{y}=\frac{1}{2}{\int }_{0}^{27}{\left(30-y\right)}^{2}-{y}^{\frac{2}{3}}dy$
${M}_{y}=\frac{22113}{5}$