Use cylindrical coordinates to evaluate ${}_{D}xydV,$ where $D$ is the solid that lies between the cylinders $x^2+y^2=1$ and $x^2+y^2=16,$ above the $xy-$plane, and below the plane $z=y+4.$

Hints: $(1){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}cossi{n}^{2}d={?}^u=sin\frac{1}{3}{u}^3+C=\frac{1}{3}si{n}^3+C$

$(2){\displaystyle \underset{0}{\overset{2}{}}}cossind={\displaystyle \underset{-}{\overset{}{}}}cossind,$ integrand is an odd function. Or use trig. identity $cossin=\frac{1}{2}sin(2).$

Set up a triple integral for the volume of the region bounded by $z=\sqrt[\phantom{2}]{3(x^2+y^2)}$ and $x^2+y^2+z^2=4$ in all three coordinate systems $($Cartesian$,$ cylindrical, spherical$).$ Evaluate the integral in spherical coordinates. Hints: $(1)$ Final answer is $\frac{8}{3}(2-\sqrt[\phantom{2}]{3}).$

$(2)$ We will talk about how to set up the limits for spherical coordinates in Tuesday's recitation.