AP Calculus AB and BC glossary

Odd function

An odd function satisfies f of negative x equals negative f of x, making its graph symmetric about the origin. Its integral over any interval symmetric about zero is exactly zero, because the two halves cancel.

f(-x) = -f(x) \implies \int_{-a}^{a} f(x)\,dx = 0

Sine, tangent and every odd power of $x$ are odd. An odd function that is defined at zero must pass through the origin, since $f(0) = -f(0)$ forces $f(0) = 0$ .

The mistake

Using the cancellation on an interval that is not symmetric. The shortcut needs limits of the form negative a to a; any other interval and the halves do not match.

Appears in: Unit 6: Integration and Accumulation