The central control of respiratory activity in the human body is exercised by the medulla oblongata, which is located in the brainstem. This region of the brain is responsible for regulating basic vital functions, including respiration. The medulla oblongata contains specialized cells called respiratory centers that receive information from sensors in the body that monitor oxygen and carbon dioxide levels. These centers then send signals to the muscles involved in breathing, instructing them to contract or relax as needed to maintain proper gas exchange.

The cerebrum, cerebellum, and hypothalamus are other parts of the brain, but they do not play a direct role in controlling respiratory activity. The cerebrum is involved in higher cognitive functions, like thinking and decision-making. The cerebellum coordinates movement and balance. The hypothalamus regulates body temperature, hormone production, and other homeostatic processes.

Therefore, the correct answer is option 3 - medulla oblongata.