The medulla oblongata (option 3) is the correct answer because it is a part of the brainstem that controls autonomic functions such as swallowing and vomiting.

Option 1, the cerebellum, is important for motor control and precision, balance, and coordination of movements but does not control swallowing or vomiting.

Option 2, the cerebrum, is the largest part of the brain and is responsible for higher brain functions such as thinking, learning, sensing, and emotion but it doesn`t regulate specific acts like swallowing and vomiting.

Option 4, the pons, is another part of the brainstem that regulates functions such as sleep, respiration, and facial expressions. However, swallowing and vomiting functions are specifically controlled by the medulla oblongata.

Thus, the medulla oblongata or option 3 is the correct answer.