We propose a type of photonic resonator with a tunable curved cavity that enables efficient tuning of the optical length of a resonant cavity made of a solid material; we call this a "tunable curved resonator" (TCR). Its integration with a "tunable curved waveguide" (TCWG) and their actuation by a MEMS (micro electromechanical systems) electrostatic comb actuator are also designed for integrated photonic circuits. With this kind of structure, a widely and continuously tunable narrow-band resonance ranging up to 200 nm is achieved with a MEMS actuation voltage less than 70 V. Its applications in widely tunable photonic filters and lasers are promising.