Suppose R is the region in the plane enclosed by y = x2 and y = 4.

Question

a) Compute the perimeter P and area A of R, and then compute the ratio Q = A/P2.
Note By squaring the perimeter the ratio becomes independent of the units chosen to measure the region.
b) Compute this ratio Q = A/P 2 for these four regions: the region R, a square, a circle, and an equilateral triangle. Draw the figures in increasing order of Q.

Answer 1

I assume you know the formula for the length of a curve in Calculus.
L = ∫( 1 + (dy/dx)^2 )^(1/2) dx (from left x to right x)

This is the hard part of the question

I ended up finding the length of the parabolic curve to be
L = 2∫(1 + 4x^2)^(1/2) dx from x=0 to x=2

I ran this through WolFram to get
http://integrals.wolfram.com/index.jsp?expr=%281%2B+4x%5E2%29%5E%281%2F2%29&random=false
sub in the values, add on the distance from (-2,4) to (2,4) and you have the perimeter of R

For the area:
A = 2∫(4 - x^2) dx from x = 0 to 2
which you should be able to do quite easily.

The length of the curve is the only difficult part of the problem.