9 FURTHER APPLICATIONS OF INTEGRATION

GROUP WORK 1, SECTION 9.1
The Bizarre Coastline
In this exercise we are going to consider the areas and coastlines of some bizarre islands, and end by
analyzing a type of shape that has intrigued mathematicians for many years. Put on your shorts and
sunglasses; we are heading to the Pacific Ocean.
1. The first island on our trip is the island of Eno. The best way to model this island is to shade the region
between the curves x = 0, x = 2π, y = 0, and y = sin x + 2.
The length of the coastline of this island is merely the sum of four arc lengths. The southern border has
length 2π, and the east and west borders each have length 2. You can compute the north border’s length
by using the arc length formula to obtain a numerical estimate accurate to three decimal places.
Coastline Length of Eno:
Area of Eno:
2. Let us reluctantly leave the island of Eno, and wander over to the neighboring island of Owt. The south,
east, and west borders of Owt look the same as Eno’s do, but the northern border is given by y = sin 2x+2.
In order to be able to deduct the cost of our trip, we need to do some research here as well. Please compute
the coastline length of Owt and the area of Owt.
Coastline Length of Owt:
Area of Owt:
488