Calculation of moments of inertia is often challenging for introductory-level physics students due to the use of integration, especially in non-Cartesian coordinates. Methods that do not employ calculus have been described for finding the rotational inertia of thin rods and other simple bodies. In this paper we use the parallel axis theorem and the perpendicular axis theorem (both of which may be proved without calculus), along with rotational symmetry, to determine, without using integration, the moments of inertia of uniform disks and spheres. (Contains 2 figures.)