Triple Integrals in Cylindrical Coordinates:

1. Evaluate  \(\displaystyle{ \iiint_E  \sqrt{x^2+y^2} \, dV} \), where \( E \) is the solid that lies inside the cylinder \(x^2+y^2=16\) and  between the planes \(z=-2\) and \( z=4\).   solution

2. Evaluate  \(\displaystyle{ \iiint_E x \, dV} \), where \( E \) is the solid that lies between the cylinders \(x^2+y^2=1\) and  \( x^2+y^2=4\), above the \(xy\)-plane, and below the plane \(z=4-x-y\).   solution

3. Rewrite the following integral in cylindrical coordinates (do not evaluate).  \(\displaystyle{ \int_{-3}^3\int_0^{\sqrt{9-x^2}}\int_0^{9-x^2-y^2} \sqrt{x^2+y^2}\,dz\,dy\,dx}\)  solution

4. Evaluate  \(\displaystyle{ \int_{-1}^1 \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} \int_{x^2+y^2}^{2-x^2-y^2} \sqrt{x^2+y^2} \,\, dz \, dy\, dx} \)   solution

5. Find the volume of the solid E that lies within both the cylinder \( x^2+y^2=1\) and the sphere \(x^2+y^2+z^2=4 \).  solution

6. Set up (do not evaluate) the triple integral whose value gives the volume of the solid E that lies between the cylinder \( x^2+y^2=1\) and the sphere \(x^2+y^2+z^2=4 \).  solution