Evaluate the triple integral $\iiint\limits_D z \ dV$ over this domain $D$ Evaluate the integral $$ \iiint \limits_D z \ dV ,$$ where $D$ is the region bounded by the planes $y = 0$, $x = 0$, $z = 0$, $z = 1$, and the cylinder $x^2+y^2=1$ with $x,y \ge 0$. 

Usually the toughest part of these problems is finding the limits of integration.  If we concentrate on just the $xy$-plane for a moment we can find the limits of integration for $x$ and $y$.  The region in the $xy$-plane over which you are integrating is the region bounded by the circle $x^2+y^2=1$ in the first quadrant.  Along the $x$-axis this region runs from $x=0$ to $x=1$.  If we pick a particular $x$, then $y$ will run from $y=0$ to $y=\sqrt{1-x^2}$.  So, now we've got bounds on $x$ and $y$.  As for $z$, that certainly runs from $z=0$ to $z=1$.  This gives all the bounds and the integral is
$$
\int_0^1\int_0^1\int_0^{\sqrt{1-x^2}}z\,dy\,dx\,dz.
$$
