Mathematics and Statistics · Ch 7 — Applications of Definite Integration

Regions Below the Axis and the Absolute-Value Rule

The formula $A=\displaystyle\int_{a}^{b} y\,dx$ was derived assuming the curve lies ABOVE the x-axis. When a curve dips below the x-axis, $y=f(x)$ is negative on that stretch, so the strips have a negative signed height and the definite integral comes out negative. Since a geometric area cannot be negative, one extra step is required.

The rule

If the region (or part of it) lies below the x-axis on $[a,b]$ , so that $f(x)\le 0$ there, the definite integral $\displaystyle\int_{a}^{b} f(x)\,dx$ is negative or zero, and the area is its absolute value:

A = \left|\int_{a}^{b} f(x)\,dx\right|

The magnitude is the correct area; the minus sign only records that the region sits below the axis.

Watch out

A curve that crosses the axis must be split

If a curve is above the x-axis on one part of the interval and below it on another (i.e. it crosses the axis inside $[a,b]$ ), the two parts partially cancel in a single integral, which then UNDER-states the true area. The correct approach is to find the crossing point, integrate each part separately, take the absolute value of each, and ADD the magnitudes:

$A = \left|\int_{a}^{c} f(x)\,dx\right| + \left|\int_{c}^{b} f(x)\,dx\right|$

where $x=c$ is the point at which the curve crosses the axis. Never let a below-axis part silently subtract from an above-axis part.

Why this matters …

Definition 1Absolute-Value Rule for Area Below the Axis

When a region lies below the x-axis, its definite integral is negative; the geometric area is the absolute value, $A=\left|\displaystyle\int_{a}^{b} f(x)\,dx\right|$ . If the curve crosses the axis inside $[a,b]$ , split at the crossing point and add t …

Definition 2Net Signed Area vs Geometric Area

A raw definite integral gives NET signed area (above-axis positive, below-axis negative). The geometric area asked for in most problems is the total size of all pieces regardless of side, obtained by t …

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