Triple Integrals in Spherical Coordinates — Question 8

Question 8 – Solution

Strategy. The distance to the $z$-axis is $\rho\sin\phi$; combine its square with the radial density and spherical Jacobian.

Step 1: Region and mass

$$M=\int_0^{2\pi}\int_0^\pi\int_1^2 \rho\,\rho^2\sin\phi\,d\rho\,d\phi\,d\theta =(2\pi)(2)\frac{15}{4}=\boxed{15\pi}.$$

Step 2: Moment of inertia $$\begin{align*} I_z&=\iiint_E(\rho\sin\phi)^2\delta\,dV\\ &=\int_0^{2\pi}\int_0^\pi\int_1^2 \rho^5\sin^3\phi\,d\rho\,d\phi\,d\theta\\ &=(2\pi)\left(\frac 43\right)\left(\frac{2^6-1}{6}\right) =\boxed{28\pi}. \end{align*}$$

Step 3: Radius of gyration $$\boxed{k_z=\sqrt{\frac{I_z}{M}}=\sqrt{\frac{28}{15}}}.$$ It lies between the minimum and maximum distances $0$ and $2$ from the axis, as required.