Area between curves... which curve is which?
When you’re doing an area between curves problem, one of the toughest parts of the problem is figuring out the orientation of the curves. And that part is actually the crux of the whole area between curves problem. Whenever you’re doing an area between curves problem, you always start by figuring out the intersection points, which is really simple because all you have to do is set the curves equal to each other. And you always finish off the problem by setting up the integral that represents the area between the curves. But once you’ve got that integral set up, you’re really home-free for the rest of the problem, because all you have to do from that point is evaluate the integral. The critical middle part is figuring out the orientation of the curves in and around those intersection points, because once you know the orientation, you can easily set up the integral.
So how do you figure out the orientation of the curves? Meaning, how do you figure out which curves is on the top and which is on the bottom, or which curve is on the right and which is on the left? The good news is that figuring out the orientation of the curves is actually not too difficult. All you need to do is pick a point on each interval (between the intersection points and/or the given endpoints of the interval), and evaluate both curves at that point. Whichever curve returns the larger value is the curve on the top. Or in the case of left-right oriented curves, whichever curve returns the larger value is the curve on the right.
Once you’ve got the orientation figured up, then you want to set up your integrand as (top curve - bottom curve), or (right curve - left curve).