Two differentiable functions r and s are defined on [a,b] with r(a)=s(a) and r(b)=s(b). Suppose r and s intersect at the additional points x1,x2,…,xk, where a<x1<x2<⋯<xk<b, and that r(x)=s(x) for every x not equal to one of these points.
(a) Explain why the quantity ∫ab∣r(x)−s(x)∣dx always equals the sum of k+1 definite integrals of the form ∫xjxj+1r(x)−s(x)dx, where x0=a and xk+1=b.
(b) Describe a procedure for calculating the exact area between the graphs of r and s on [a,b] without using the absolute value notation in the integrand. Be sure to state how one decides which integrand to subtract from which on each sub-interval.
(c) Give one example of a pair of elementary functions that intersect at more than two points, and briefly outline how your procedure from part (b) would apply to your example.