Consider the periodic functions and on the interval

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Integrals · Applications of Integrals

Problem

Consider the periodic functions p(x)=sin⁡xp(x)=\sin x and q(x)=sin⁡2xq(x)=\sin^{2}x on the interval [−π,π][-\pi,\pi]. The graphs of pp and qq intersect at several points and bound multiple regions of equal area. (a) Determine all xx-values in [−π,π][-\pi,\pi] where sin⁡x=sin⁡2x\sin x=\sin^{2}x. Show your work. (b) Provide a clear, labeled sketch description of the graphs of pp and qq on [−π,π][-\pi,\pi], indicating which function is above the other on each sub-interval between consecutive intersection points. (c) Develop a definite integral expression whose value equals the combined area of all regions bounded by the graphs of pp and qq on [−π,π][-\pi,\pi]. Explain any simplifications you make in setting up the integral(s). (d) Evaluate the expression from part (c) to obtain the exact total area. (e) Generalize your result by deriving a formula for the total area of the regions bounded by sin⁡x\sin x and sin⁡2x\sin^{2}x on the interval [−nπ, nπ][-n\pi,\,n\pi], where nn is any positive integer. Provide justification for your generalization.

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