Area Between Curves

Finding Enclosed Area

The area between two curves generalizes the area under a curve by integrating thin vertical strips across the region. If $f(x) \ge g(x)$ on $[a, b]$, the trapped area is:

$$A = \int_a^b \big(f(x) - g(x)\big)\, dx$$

The key rule is top minus bottom: the upper curve minus the lower one, ensuring the strip height is nonnegative. Unlike a single definite integral, which computes signed area, between-curves area is always positive because you subtract the lower function from the upper, making the $x$-axis completely irrelevant.

Pitfalls and Alternatives

A major common pitfall is integrating without checking curve order or ignoring crossings. If $f$ and $g$ cross inside $[a, b]$, you must split the integral at the crossing point and compute top-minus-bottom on each sub-interval separately. Subtracting in the wrong order yields a negative area.

When curves are functions of $y$, integrate horizontally using right minus left strips.