The region R is bounded by $\displaystyle z=0$, $\displaystyle z=\frac{1}{\sqrt{x^{2}+y^{2}+1}}$(in blue), $\displaystyle y=0,$ and $y=\sin\!\left(x\right)$(in red). To find the centroid of the region you would compute $\displaystyle\int\int\int_R\delta(x,y,z) \,dV= \int_a^b\int_{c}^{d}\int_e^f dz\,dy\,dx, \int_a^b\int_{c}^{d}\int_e^f x\,dz\,dy\,dx,\int_a^b\int_{c}^{d}\int_e^f y\,dz\,dy\,dx$ and $\displaystyle\int_a^b\int_{c}^{d}\int_e^f z\,dz\,dy\,dx$ where
$a =$
$b =$
$c =$
$d =$
$e =$
$f =$
and finally the centroid is (a computer may be helpful at this point)
$\overline {x} =$
$\overline {y} =$
$\overline {z} =$