Finding the Area Between Curves Expressed as Functions of y

Some regions are a nightmare to slice vertically. If the boundary curves double back on themselves, a single x value can hit the region twice, so there's no clean top function and bottom function. But solve each curve for x in terms of y instead, and the trouble often disappears: for each height y, the region runs from one clear left boundary to one clear right boundary. That's your cue to slice horizontally.

A horizontal sliver has width dy and length equal to x_right(y) minus x_left(y). Stack these slivers from the lowest y value to the highest, and the integral of (x_right - x_left) dy adds them up into the total area — same logic as top minus bottom, just rotated ninety degrees. Find the limits by setting x_right(y) = x_left(y) and solving for the y values where the curves cross.

This is not just swapping letters in your old setup. You have to actually re-solve each curve as x = g(y), decide which one sits to the right, and reset your limits as y values instead of x values. Done right, though, it gives the exact same number as slicing the same region vertically in x — area doesn't care which way you cut it. Choosing the easier direction is really the whole skill: it's the same region, the same area, found by whichever slicing avoids splitting the integral into pieces.