Arc Length Integral Questions

Question

Let R be the region in the first quadrant bounded by y=x² and y=√x. Write down a single integral that represents the perimeter of R.

Answer 1

Since the functions are inverses of each other over [0,1], you can double the length of either curve.

So a possible answer:

2 ∫¹₀√(4x²+1) dx, where (2x)² is the derivative of x² squared.

The following graph proves the point and shows how to evaluate the integral (using trig substitution).

desmos.com/calculator/kxx8btzuzs

Answer 2

The limits of integration are 0 to 1.

The integrand is {(1+4x²)^1/2+[1+(x/4)]^1/2} dx