Area Between Two Curves

Definite integration also measures the area of a region trapped between two curves, rather than between one curve and an axis.

The formula

Let $y=f(x)$ and $y=g(x)$ be two continuous curves with $f(x)\ge g(x)$ (so the curve $f$ is the UPPER boundary and $g$ is the LOWER boundary) on the interval $a\le x\le b$. The area of the region enclosed between them is

Each thin vertical strip now runs from the lower curve up to the upper curve, so its height is the DIFFERENCE $f(x)-g(x)$ and its area is $[f(x)-g(x)]\,dx$. Integrating these strip areas from $x=a$ to $x=b$ gives the enclosed region.

Finding the limits — the points of intersection

When the region is bounded only by the two curves (not by given ordinates), the limits $a$ and $b$ are the x-coordinates of the points where the curves meet. These are found by solving $f(x)=g(x)$ — that is, setting the two equations equal and solving for $x$. The smaller root is the lower limit $a$, the larger root is the upper limit $b$.

The method

1. Find the intersection points by solving $f(x)=g(x)$; these give the limits $a$ and $b$.
2. Decide which curve is on top on that interval (test a value between $a$ and $b$, or reason from the shapes). The upper curve is $f$, the lower is $g$.
3. Integrate the difference: $A=\displaystyle\int_{a}^{b}[f(x)-g(x)]\,dx$.