MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Two differentiable functions f and g are defined for −3≤x≤3. The function f is even, so f(−x)=f(x) for all x in the domain, while g is odd, so g(−x)=−g(x). The graphs of f and g intersect exactly at x=−2,0,2, creating several closed regions between the two graphs on the interval [−3,3].
(a) Describe qualitatively how the parity of f and g affects the arrangement and shape of the regions bounded by the two graphs on the sub-intervals [−3,−2], [−2,0], [0,2], and [2,3]. Address which regions are mirror images of others and which lie above or below the x-axis.
(b) A student evaluates 2∫02∣f(x)−g(x)∣dx to obtain the total area of every bounded region between the graphs on [−3,3]. Without performing any numerical integration, explain whether the student’s procedure is always guaranteed to produce the correct total area. In your explanation cite specific consequences of the evenness of f, the oddness of g, and the given intersection points.