In-class work for 6 July 2012

1. Evaluate the integral using cylindrical coordinates
    a.) $\displaystyle\int\int\int_E \sqrt{x^2+y^2} dV$ where $E$ is the region that lies inside the cylinder $x^2+y^2=16$ and between the planes $z=-5$ and $z=4$.
    b.) $\displaystyle\int\int\int_E x dV$ where $E$ enclosed by the plane $z=0$ and $z=x+y+5$ and by the cylinders $x^2+y^2=4$ and $x^2+y^2=9$.

2. Find the volume of the solid that lies within both the cylinder $x^2+y^2=1$ and the sphere $x^2+y^2+z^2=4$.

3. Evaluate the integral by changing to cylindrical coordinates
    a.) $\displaystyle\int_{-2}^2 \int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}} \int_{\sqrt{x^2+y^2}}^2 xz dz dx dy$
    b.) $\displaystyle\int_{-3}^3 \int_0^{\sqrt{9-x^2}} \int_0^{9-x^2-y^2} \sqrt{x^2+y^2} dz dy dx$.