Before completing this research task, I think it is important to think about the relevance of exploration today. This task was inspired by my visit to Brewster Glacier on the West Coast of New Zealand, which you can read about here. Fortunately, despite the fact we can see these glaciers from satellites, there is a need for constant surveillance to document how they change over time. Check out this time lapse of the Franz Josef Glacier.

But, on a more human level, there is a sense of awe, wonder, and beauty that makes these places worth visiting. They put our existence into the perspective and remind us of a time before humans had made their presence known on this Earth. For this reason, although the mathematical questions are interesting, it is important to recognize the importance of experience and the humanities to make us care about these places. Anyway, get back to the math.

1. Find the area and perimeter of the figure below and make sure to explain your procedure:
2. Find the area and perimeter of the figure below and explain your process:
3. The curve above can be broken into a series of smaller and smaller rectangles to get an increasingly accurate area. Investigate summation notation and explain how the expression below results in a good approximation:
4. Rather than the approximation offered by the previously summation, integrals offer a perfect calculation of area. Investigate what an integral is and explain how the expression below results in the perfect area:
5. Find the area and perimeter of the figure below and explain your process:
6. How does your answer and/or strategy change if I give you the equations for the curves that make up the figure?
7. Check out this link to Brewster Glacier on Google Earth (GE), where you can use the ruler icon on the left-hand menu to find the perimeter and area of the glacier: You’ll probably notice that GE does not simply give you the area and perimeter. How does it complete its calculations and what are the implications?
8. Click here in order to access a Desmos plot with Brewster pasted in. Think about problems 1-6 and identify, implement, and reflect on at least 3 ways to find the area and perimeter of the glacier. Make sure you identify the strengths and flaws of each approach. Also, if you took any incorrect approaches, include a discussion of those as well.