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

A vertical strip at $x$ has height $y_{\text{top}} - y_{\text{bottom}}$ and width $dx$, so the area between two curves is $\int_a^b (f - g)\,dx$ with $f$ on top and with $x$-values as the limits. Nothing assumes either curve is above the axis, since a vertical distance does not depend on where the axis sits. Unless the interval is given, the limits come from solving $f(x) = g(x)$ for every root: $x^{2} = x + 2$ gives $x = -1$ and $x = 2$, and $y = x^{2}$ with $y = 4$ gives $x = -2$ and $x = 2$, so stopping at one root or importing a number from the prompt loses part of the region.

Which curve is on top is settled by testing a value strictly between consecutive intersections, not by comparing the formulas: on $[0, 2]$ the line $y = 2x$ lies above $y = x^{2}$, even though the parabola overtakes it for every $x > 2$, and a bounded region is precisely where the eventual ordering is reversed. A negative answer means the subtraction ran backwards, and the repair is to swap the order rather than to take an absolute value at the end. That distinction has no consequences while the region is one piece and becomes essential in 8.6, where $\left|\int (f - g)\right|$ and $\int |f - g|$ stop agreeing.