Energy density of any capacitor or inductor

Most introductory physics courses derive the energy densities of the static electric and magnetic field for the simple cases of parallel plate capacitors and infinitely long solenoids. Authors then inform readers, usually without proof, that the energy density equations derived for the simple cases are actually the correct equations for all capacitors and inductors, regardless of their shape. The proof of the general case is typically omitted because it involves the differential calculus of vector fields. In view of this, we provide a derivation for the energy density only based on integral calculus for capacitors and inductors of any kind. The derivation, albeit seemingly complicated at first, is conceptually simple enough for introductory physics courses and does not require any knowledge of the differential calculus of vector fields.