The area enclosed between two curves

The most useful case is not the area under a curve, but the area between two curves. The formula is remarkably simple: you integrate the difference of the two functions.

If, on $[a, b]$, the curve of $f$ is above that of $g$, that is, $f(x) \geq g(x)$, the area of the region enclosed between the two is:

$$\mathcal{A} = \int_a^b \bigl(f(x) - g(x)\bigr)\,\mathrm{d}x$$

that is, the top one minus the bottom one.

The remarkable point: this formula remains valid even if the curves go below the axis. Their position relative to the x-axis does not matter — only their position relative to each other counts. The difference top minus bottom is always positive, so the area is too. You no longer have to worry about the sign of $f$ or of $g$ separately, only about knowing which one is on top.

Example. Between the line $y = x$, on top, and the parabola $y = x^2$, below, on $[0, 1]$:

$$\mathcal{A} = \int_0^1 \left(x - x^2\right)\mathrm{d}x = \left[\frac{x^2}{2} - \frac{x^3}{3}\right]_0^1 = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}$$

Simple, robust, and free of sign problems. All that remains is to find the bounds.