Finding the Area Between Curves Expressed as Functions of y AB & BC

A horizontal strip at height $y$ has length $x_{\text{right}} - x_{\text{left}}$ and thickness $dy$, so the area is $\int_c^d (x_{\text{right}} - x_{\text{left}})\,dy$ with $y$-values as the limits. This is 8.4 rotated: top becomes right, $dx$ becomes $dy$, and every check carries over unchanged. The region between $x = y + 2$ and $x = y^{2}$ is the reflection of 8.4's region across $y = x$ and has the same area $\frac{9}{2}$, found from $y^{2} = y + 2$ with $y = -1$ and $y = 2$ and a test at $y = 0$ that puts the line on the right.

Every boundary must be rewritten as $x = g(y)$ before integrating, the branch restriction travels with it, and the limits have to be recomputed as the $y$-coordinates of the intersection points rather than reused from a $dx$ setup. A surviving $x$ anywhere in a $dy$ integral means a boundary was never converted. Which variable to sweep in is decided by how often a boundary hands off to a different curve: the region between $x = y^{2}$ and $x = y + 2$ needs one integral in $y$ and two in $x$, since the upper boundary changes from the parabola's upper branch to the line at $x = 1$.