Question $1(10$ marks$)$

Evaluate the integral by changing to cylindrical coordinates:

${\displaystyle \underset{0}{\overset{1}{}}}{\displaystyle \underset{0}{\overset{\sqrt[\phantom{2}]{1-y^2}}{}}}{\displaystyle \underset{x^2+y^2}{\overset{\sqrt[\phantom{2}]{x^2+y^2}}{}}}xyzdzdxdy$

Question $2(10$ marks$)$

By using cylindrical coordinates, find the volume of the solid that the cylinder $r=3cos$ cuts out of the sphere of radius $3$ centered at the origin.

$($Hint: ${\displaystyle \underset{-\frac{}{2}}{\overset{\frac{}{2}}{}}}si{n}^{3}d=2{\displaystyle \underset{0}{\overset{\frac{}{2}}{}}}(\frac{3}{4}sin-\frac{1}{4}sin3)d)$

Question $3(8$ marks$)$

Let $H$ be a solid upper hemisphere of radius a whose density, $(x,y,z)=K\sqrt[\phantom{2}]{x^2+y^2+z^2},$ at any point is proportional to its distance from the center of the base $(0,0,0).$ Find the mass, $m,$ of it by using spherical coordinates.

$($Hint: $m=dV)$