Area Under A Curve 61

1. Core Concept Overview & Intuition

The definite integral, developed through the Fundamental Theorem of the Integral Calculus, provides a powerful tool for measuring accumulated change. When the integrand $f(x)$ is non-negative on an interval $[a,b]$, the definite integral $\int_{a}^{b}f(x)\,dx$ admits a direct geometric interpretation: it equals the area of the region bounded by the curve $y=f(x)$, the $x$-axis, and the vertical lines $x=a$ and $x=b$. This is the central idea of the present lesson.

The intuition rests on a limiting process. Suppose the interval $[a,b]$ is partitioned into thin vertical strips of width $\Delta x$. Each strip is approximately a rectangle of height $f(x)$ and width $\Delta x$, so its area is approximately $f(x)\Delta x$. Summing over all strips and letting the widths shrink to zero produces the definite integral. Thus the area is the limit of a sum of rectangular areas, and the definite integral evaluates that limit exactly.

Two refinements are essential. First, when the curve lies below the $x$-axis, the function values $f(x)$ are negative, so the definite integral returns a negative number whose magnitude equals the geometric area. The geometric area must therefore be recovered by taking the absolute value, or by integrating $-f(x)$. Second, when a curve crosses the $x$-axis within $[a,b]$, the integral computes a signed (net) area in which regions below the axis cancel regions above it. To obtain the total geometric area, the interval must be split at each zero of $f$ and the absolute values of the separate integrals added. Finally, when two curves intersect, the area of the region enclosed between them is obtained by integrating the difference of the upper and lower functions across the interval determined by their points of intersection.

2. Mathematical Formalization & Core Definitions

2.1 Objectives of the Lesson

> By the end of this lesson the learner should be able to:
> - state the formula for the area under a curve bounded by the $x$-axis and two ordinates;
> - compute the area of a region lying below the $x$-axis and interpret the sign of the definite integral;
> - compute the combined area of a region in which the curve crosses the $x$-axis;
> - determine the area of the region enclosed between two intersecting curves.

2.2 Formula for the Area under a Curve

> Definition (Area under a curve). Let $f$ be a continuous function with $f(x)\ge 0$ on $[a,b]$. The area $A$ of the region bounded by the curve $y=f(x)$, the $x$-axis, and the ordinates $x=a$ and $x=b$ is given by

$$
A=\int_{a}^{b}f(x)\,dx.
$$

Here $y=f(x)$ is the equation of the curve, $x$ is the independent variable measured along the horizontal axis, $a$ and $b$ are the limits of integration (the $x$-coordinates of the bounding ordinates), and $f(x)\,dx$ is the element of area of a representative vertical strip. The formula is a direct consequence of the Fundamental Theorem of the Integral Calculus: if $F$ is any antiderivative of $f$, then

$$
A=\int_{a}^{b}f(x)\,dx=F(b)-F(a).
$$

2.3 Area Below the $x$-axis

> Definition (Signed area). If $f(x)\le 0$ on $[a,b]$, the curve lies entirely below the $x$-axis and the definite integral is negative:

$$
\int_{a}^{b}f(x)\,dx\le 0.
$$

> The geometric area of the region is the absolute value of this integral,

$$
A=\left\lvert\int_{a}^{b}f(x)\,dx\right\rvert=-\int_{a}^{b}f(x)\,dx.
$$

The negative sign arises because each element $f(x)\,dx$ is negative when $f(x)<0$. Geometrically, the region lies beneath the axis, and its area is a positive quantity obtained by integrating the positive function $-f(x)$, which is the reflection of $f$ in the $x$-axis.

2.4 Combined Area under a Curve

> Definition (Combined area). Suppose $f$ is continuous on $[a,b]$ and changes sign at interior points $c_1,c_2,\dots,c_n$ with $a<c_1<c_2<\cdots<c_n<b$. The total geometric area of the region between the curve and the $x$-axis is

$$
A=\int_{a}^{c_1}\lvert f(x)\rvert\,dx+\int_{c_1}^{c_2}\lvert f(x)\rvert\,dx+\cdots+\int_{c_n}^{b}\lvert f(x)\rvert\,dx.
$$

In practice each sub-integral is evaluated separately and its absolute value taken, and the resulting positive areas are added. The points $c_i$ are precisely the zeros of $f$ in $(a,b)$, that is, the points where the curve meets the $x$-axis.

2.5 Area Between Two Intersecting Curves

> Definition (Area between two curves). Let $y=f(x)$ and $y=g(x)$ be continuous on $[a,b]$ with $f(x)\ge g(x)$ there. The area of the region bounded by the two curves between $x=a$ and $x=b$ is

$$
A=\int_{a}^{b}\bigl[f(x)-g(x)\bigr]\,dx.
$$

When the curves intersect, the limits $a$ and $b$ are the $x$-coordinates of the points of intersection, obtained by solving $f(x)=g(x)$. If the curves cross one another inside the interval, the interval is subdivided at the crossing points and the absolute difference is integrated on each subinterval, so that

$$
A=\int_{a}^{c}\bigl\lvert f(x)-g(x)\bigr\rvert\,dx+\int_{c}^{b}\bigl\lvert f(x)-g(x)\bigr\rvert\,dx,
$$

where $c$ is an interior intersection point. The integrand $f(x)-g(x)$ is the height of a representative vertical strip measured from the lower curve to the upper curve.