A continuous function is defined on the closed interval with
MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change
Problem
A continuous function f is defined on the closed interval [−a,a] with a>0. It is known that f is odd, that is, f(−x)=−f(x) for every x in the domain.
(a) Describe how the symmetry of an odd function on [−a,a] affects the signed areas represented by ∫−a0f(x)dx and ∫0af(x)dx. Provide a verbal explanation without evaluating any integrals.
(b) Using your reasoning from part (a), determine the average value of f on [−a,a]. Justify your answer algebraically in terms of the definition of average value.