Final answer:
The integumentary system covers the external surface of the body but not the internal surface of the mouth. It includes the skin and its associated structures, and plays a vital role in protecting the body and maintaining homeostasis.
Step-by-step explanation:
The organ system that covers the external surface of the body but not the internal surface of the mouth is the integumentary system. This system includes the skin and its associated structures, such as hair and nails. The skin acts as a protective barrier, maintaining homeostasis and preventing harmful microorganisms and toxins from entering the body.