Mathematics · Ch 7 — Integrals

Area Function

7.8.1

Area Function

The Area Function: A Bridge from Definite Integral to Antiderivative

The definite integral $\int_a^b f(x) \, dx$ is the area of the region bounded by the curve $y = f(x)$ , the $x$ -axis, and the vertical lines $x = a$ and $x = b$ — a fixed number for given $a$ and $b$ . But what happens if we let the upper limit vary?

Introducing the Area Function $A(x)$

For a point $x$ in $[a, b]$ , look at the definite integral from the fixed lower limit $a$ up to the variable point $x$ :

\int_a^x f(t) \, dt

(We use $t$ as the dummy variable of integration to avoid confusion with the upper limit $x$ .) This area is no longer constant — it changes as $x$ changes, so it is a function of $x$ , denoted $A(x)$ and called the Area Function.

A(x) = \int_a^x f(t) \, dt

We assume $f(x) > 0$ on $[a, b]$ so the area is positive, though the theorems that follow hold for any integrable function.

The Role of the Area Function

$A(x)$ is the conceptual link between integration (finding area) and differentiation (finding rate of change). The two Fundamental Theorems of Calculus are built directly on it:

First Fundamental Theorem: the derivative of $A(x)$ is the original function: $A'(x) = f(x)$ . So $A(x)$ is an antiderivative of $f(x)$ .
Second Fundamental Theorem: for any antiderivative $F$ of $f$ , $\int_a^b f(x) \, dx = F(b) - F(a)$ . …

Figure 7.1Area function A(x) under the curve y = f(x) on [a, b] (Fundamental Theorem of Calculus)

Fig. 7.1 — Area function A(x) under the curve y = f(x) on [a, b] (Fundamental Theorem of Calculus)

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What the Figure Shows

The diagram is a standard Cartesian plot with the origin at $O$ . The horizontal axis is labelled $x$ and the vertical axis $y$ . A smooth, increasing curve $y = f(x)$ is drawn entirely in the first quadrant, above the $x$ -axis. Three points are marked on the $x$ -axis: $a$ , $x$ , and $b$ , with $a < x < b$ . From each of these three points, a vertical dashed line (an ordinate) rises straight up to meet the curve at the points $(a, f(a))$ , $(x, f(x))$ , and $(b, f(b))$ .

The region between the curve, the $x$ -axis, and the two ordinates at $x = a$ and $x = x$ is shaded in a lighter tint. This region is labelled $A(x)$ . The adjacent region between $x = x$ and $x = b$ is shaded in a darker tint. The key visual idea is that the lighter region's area changes as the point $x$ slides along the $x$ -axis — the area is a function of $x$ .

The Physical Idea

The figure teaches a fundamental shift in perspective. Instead of thinking of a definite integral $\int_a^b f(t)\,dt$ as a single fixed number (the total area from $a$ to $b$ ), we treat the upper limit as a variable. For any $x$ in $[a, b]$ , the integral from the fixed left endpoint $a$ up to that variable $x$ gives a running total of area. This running total is itself a function of $x$ , called the area function $A(x)$ .

The lighter shaded region represents $A(x)$ — the area accumulated so far, starting from $a$ and stopping at $x$ . The darker region is the remaining area from $x$ to $b$ , which is $\int_x^b f(t)\,dt$ . As $x$ moves to the right, the lighter region grows and the darker region shrinks. The total area from $a$ to $b$ is the sum of the two: $A(b) = \int_a^b f(t)\,dt$ .

The Key Formula

The textbook defines the area function as

A(x) = \int_a^x f(t)\,dt

Here:

$A(x)$ is the area of the lighter shaded region (a function of $x$ ).
$a$ is the fixed left endpoint of the interval.
$x$ is the variable upper limit, with $a \le x \le b$ .
$f(t)$ is the height of the curve at a general point $t$ between $a$ and $x$ .
The dummy variable $t$ is used inside the integral to avoid confusion with the upper limit $x$ .

The crucial result that follows from this figure — and which the textbook states as the First Fundamental Theorem of Calculus — is that the derivative of this area function gives back the original function:

Important

$A'(x) = f(x)$

The rate at which the shaded area grows, as $x$ increases, is exactly the height of the curve at that point.

Note

The figure assumes $f(x) > 0$ on $[a, b]$ so that area is positive and the visual is clear. However, the theorem holds for any integrable function — if $f(x)$ is negative, the "area" becomes signed (below the $x$ -axis), and the derivative result $A'(x) = f(x)$ remains valid.

Why This Matters …

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Area Function

Area Function

The Area Function: A Bridge from Definite Integral to Antiderivative

Introducing the Area Function A(x)A(x)A(x)

The Role of the Area Function

What the Figure Shows

The Physical Idea

The Key Formula

Why This Matters …

Introducing the Area Function $A(x)$