In this activity, we will work with a few triple integrals and look at the advantages of utilizing cylindrical coordinates for the iterated integral calculation.
Suppose the density of the cone defined by \(r = 1 - z\text{,}\) with \(z \geq 0\text{,}\) is given by \(\delta(r, \theta, z) = z\text{.}\) A picture of the cone is shown at in Figure 12.7.2. Set up an iterated integral in cylindrical coordinates that gives the mass of the cone. You do not need to evaluate this integral.
Write an iterated integral expression in cylindrical coordinates whose value is the volume of the solid bounded below by the cone \(z = \sqrt{x^2+y^2}\) and above by the cone \(z = 4 - \sqrt{x^2+y^2}\text{.}\) A picture is shown in Figure 12.7.3. You do not need to evaluate this integral.
