Describe a specific example of how the integumentary system helps maintain homeostasis.

In a word? Sweat.

"Integumentary system" and "homeostasis" are big scary words, but what we're really talking about here is how skin and the organ system it is a part of help the human body regulate itself, particularly with regard to temperature and moisture.

Specialized nerves in the skin are sensitive to changes in temperature; specifically, certain nerves are sensitive to high temperatures while others are sensitive to low temperatures.

If high-temperature nerves are triggered, they send a signal to the brain (chiefly the brain stem and the hypothalamus) that activates the perspiration response, telling sweat glands in the skin to open up and release more perspiration that will conduct heat away from the skin and evaporate away. The sweat response can also be activated simply to reduce excess moisture, but usually that's handled by the urinary system.

If low-temperature nerves are triggered, they send a signal to the brain that instead activates the shivering response, telling muscles across the body to vibrate rapidly in a way that will generate heat.

The skin also provides passive protection against loss of heat and moisture, by layers of fat and keratin that block heat transfer and evaporation.

Finally, human skin also does a process comparable to photosynthesis for the creation of vitamin D; the reason people have different skin colors is that humans who evolved in regions of high sunlight had a natural selection pressure for dark skin to protect against radiation damage (especially to folate) while humans who evolved in regions of low sunlight had a natural selection pressure for light skin to maximize the production of vitamin D. Since folate and vitamin D are both important in pregnancy, the selection pressure was quite strong and this evolutionary process was quite fast, occurring over only a few tens of thousands of years.