Who is asking: Student
I was looking at the relationship of the area of a circle and the circumference when I realized that 2*pi*r is the derivative of pi*r^2. I was wondering if there is any connective deep dark meaning as to why this appears to be related. Thanks for any help you can give me!
You can also do this for a square - provided you set up the origin as the center of the square and x as half the length of a side. The layers are smaller squares of perimeter 8 h at distance h from the center. The area of thin square ring is 8 h and the total area is then - the actual area of a square with sides 2x.
Try it with some other shapes. The key is a consistent breaking up of the object into thin strips of thickness . Integrating the lengths of these strips from the smallest to the largest, with respect to h then gives the area.
This connection is nice for reducing memory work: remember one formula and get the other by differentiation or integration! It also leads on to some important insights and methods of integration as you learn more calculus.
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