Let R be the region in the upper half plane between the curves and .
Find:
-Dan
Let's start by looking at the hint and describe the curves.
1)
2)
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].
Thus
The integration is now simple. I'll sketch it briefly:
-Dan
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.