AP Calculus AB and BC glossary

Total area

Also called: Unsigned area

Every piece of a region counts positively in total area, so the interval is split at each sign change and the pieces where the function is negative are negated before being added. In symbols it is the integral of the absolute value of f, and between two curves the integral of the absolute value of f minus g.

∫ab∣f(x)∣ dx\int_a^b \bigl|f(x)\bigr|\,dx

How total area differs from net signed area is set out on the area vs net signed area page; what matters here is the bookkeeping. Every sign change is a break point: the zeros of ff, plus any jump in a piecewise definition where ff switches sign without passing through zero. On each piece ff keeps one sign, so the absolute value comes off and the piece can be added by size.

∫ab∣f(x)∣ dx=∣∫ac1f∣+∣∫c1c2f∣+∣∫c2bf∣\int_a^b \bigl|f(x)\bigr|\,dx=\left|\int_a^{c_1} f\right|+\left|\int_{c_1}^{c_2} f\right|+\left|\int_{c_2}^{b} f\right|

A region between two curves works the same way on the difference: its total area is ∫ab∣f(x)−g(x)∣ dx\int_a^b |f(x)-g(x)|\,dx, split at every intersection, because that is where the top curve and the bottom curve trade places. Comparing the two integrals always gives ∣∫abf∣≤∫ab∣f∣\left|\int_a^b f\right| \le \int_a^b |f|, with equality exactly when ff does not change sign on [a,b][a,b].

The mistake

Writing ∫ab(∣f(x)∣−∣g(x)∣) dx\int_a^b \bigl(|f(x)|-|g(x)|\bigr)\,dx for the area between two curves. The absolute value belongs around the difference, not around each curve separately. Where f=1f=1 and g=−3g=-3 the vertical gap is 44, while ∣f∣−∣g∣|f|-|g| reports 1−3=−21-3=-2.

Appears in: Unit 6: Integration and Accumulation, Unit 8: Applications of Integration