Topic 8.4 Finding the Area Between Curves Expressed as Functions of x: integrate the top minus the bottom curve to find the enclosed area.

What this topic is asking

The College Board (Topic 8.4) finds the area between two curves that are functions of $x$. The area is the integral of the upper function minus the lower function over the interval between their intersection points.

The method

A worked area

Why top minus bottom works regardless of the axis

The height of a thin vertical strip between the curves at a given $x$ is always (upper $y$) minus (lower $y$), whatever the signs of those $y$-values. If both curves are below the $x$-axis, the difference of two negatives is still the correct positive strip height; if the region straddles the axis, the subtraction handles it automatically. This is why area-between-curves never needs the curves to be above the axis and never needs absolute values, provided you have correctly identified which curve is upper. The single subtraction captures the geometry.

Finding intersections and the top curve carefully

Two preliminary steps cause most errors. First, the limits of integration are the intersection $x$-values, found by solving $f(x) = g(x)$; using the wrong limits (such as the $x$-intercepts of one curve) gives the wrong region. Second, you must confirm which curve is on top by testing a point in the interval, because integrating bottom minus top gives a negative of the area. When the curves are simple and one clearly dominates, this is quick, but on the no-calculator section it is worth one explicit test. With the right limits and the right top curve, the integral of top minus bottom gives the area directly.

The vertical-strip picture

The formula follows from slicing the region into thin vertical strips. At a position $x$, a strip runs from the lower curve $g(x)$ up to the upper curve $f(x)$, so its height is $f(x) - g(x)$ and its area is that height times the width $dx$. Summing the strips from $x = a$ to $x = b$ gives $\int_a^b [f(x) - g(x)]\,dx$. This picture explains why you integrate in $x$ when the curves are functions of $x$: the strips are vertical, indexed by $x$, and the height is read off as top minus bottom. Holding the strip image makes the setup automatic and shows immediately why the wrong subtraction order would give a negative, unphysical strip height.

When one boundary is an axis

Often one of the two "curves" is simply the $x$-axis, $y = 0$, so the area between a curve and the axis is $\int_a^b f(x)\,dx$ where $f \ge 0$. If the curve dips below the axis on part of the interval, that portion contributes a negative signed integral, so the area there is $\int |f(x)|\,dx$, computed by splitting at the $x$-intercepts and taking magnitudes (the multiple-intersection idea). Treating the axis as the lower curve $g(x) = 0$ folds this into the same top-minus-bottom framework, with the reminder that area requires the curve to stay above the chosen lower boundary or else the region must be split.