Hint: Is there a relationship between the two boundary curves?
Let's start by looking at the hint and describe the curves.
Notice that equation 2) can be rewritten as so equation 2) is just equation 1) with a vertical stretch factor of 2.
I am going to represent equation 1) as y = f(x) and equation 2) as y = 2 f(x).
A couple of important facts about curve 1). Note that f(-x) = f(x), so f(x) is an even function. Also note that f(x) has two real zeros: . And finally, if we look at y = f(x) in quadrant 1 we find that f(x) is monotonically decreasing on [0, 1], so f(x) is one to one in Quadrant 1. (And thus is a function on [-1, 1].) Similar comments obviously apply to y = 2 f(x), and it is trivial to show that the only intersection points of equations 1) and 2) are at .
Our region of integration can thus be characterized as (2 f(x) - f(x))dx integrated over the interval [-1, 1].
The integration is now simple. I'll sketch it briefly:
There are four points of intersection:
Does your solution account for all four points of intersection?
Edit: Never mind. I missed the part that said that is restricted to the upper half-plane.