The geometric area of a region

To get the true geometric area of a surface — the one you would shade in, always positive — you must correct the sign where the function goes below the axis.

The method is systematic:

1. Find where $f$ changes sign, by solving $f(x) = 0$.
2. Split the interval into pieces on which $f$ keeps a constant sign.
3. On each piece, take the absolute value of the integral.
4. Add up these positive areas.

In other words, the geometric area between the curve and the axis is written:

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

In practice, you don't compute an antiderivative of $|f|$, which would be difficult: you split according to the sign. Take $f$ on $[a, b]$, positive on $[a, c]$ then negative on $[c, b]$:

$$\mathcal{A} = \int_a^c f(x)\,\mathrm{d}x - \int_c^b f(x)\,\mathrm{d}x$$

The first piece is already positive. The second, where $f$ is negative, has a negative integral: by subtracting it, you make it positive. The two contributions then add up like real areas.

Notion	Formula	Sign
signed area	$\displaystyle\int_a^b f$	can be negative or zero
geometric area	$\displaystyle\int_a^b \lvert f \rvert$, by splitting	always positive

The reflex to keep: before computing an area, study the sign of $f$. If it changes, split. That is what separates a correct area from a wrong result.