Although the research of tunable THz absorbers has attracted considerable attention, improving the multifunctional switching of the absorbers to fulfill more application scenarios is still worth studying. This paper proposes an integrated design of a dynamically switchable multi-narrowband and wideband absorber based on graphene and vanadium dioxide(VO2) in the terahertz region through voltage and thermal manipulation. A compact structure with five layers, from top to bottom, namely, graphene layer, dielectric layer, vanadium dioxide-metal layer, dielectric layer, and metal reflective layer, is proposed. Simulation results show that the absorption of the broadband absorber exceeds 90% in the frequency range of 2.44–4.48 THz when the VO2 is in the metallic state and the Fermi energy level of graphene is 1 eV. Meanwhile, a wide-band reflector with a reflection rate of more than 80% in the entire THz frequency range of 0–10 THz is shown. When adjusting VO2 to a dielectric state, while the Fermi energy level of graphene is controlled at 0.7 eV, a multi-narrowband absorber with four frequency points, respectively at 3.41 THz, 6.66 THz, 8.9 THz, and 9.79 THz, and three points of high frequency have an absorption rate close to 100%. In addition, we analyze the physical mechanism of the proposed absorber by impedance matching theory and electric and magnetic field distribution, and the robustness of the angle of incident wave. Due to its switchable function, the proposed metamaterial absorber has great potential for applications in terahertz switching, sensing, and electromagnetic stealth.