Let and for all real numbers

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

Let p(x)=cos⁡xp(x)=\cos x and q(x)=12q(x)=\dfrac12 for all real numbers xx. Focus on the interval [−π,3π][-\pi,3\pi]. (a) Justify that the graphs of y=p(x)y=p(x) and y=q(x)y=q(x) intersect at more than two distinct points in the interval [−π,3π][-\pi,3\pi]. State those xx–coordinates explicitly. (b) Without solving any trigonometric equations other than those in part (a), set up—but do not evaluate—an integral whose value equals the total area of the regions bounded by the two graphs on [−π,3π][-\pi,3\pi]. Your integrand should be written as a single elementary expression in xx. (c) Evaluate the integral from part (b) exactly. Show all antiderivative work and simplify your final answer.

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