Between and , the graphs of and intersect twice and enclose a finite region. This…

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

Problem

Between 00 and 2π2\pi, the graphs of y=sin⁡xy=\sin x and y=cos⁡xy=\cos x intersect twice and enclose a finite region. This region is revolved about the horizontal line y=−2y=-2. Answer the following questions. Exact answers are required unless otherwise indicated. (a) Find all values of xx in the interval [0,2π][0,2\pi] for which sin⁡x=cos⁡x\sin x=\cos x. (b) Identify the specific subinterval of xx over which sin⁡x\sin x and cos⁡x\cos x enclose a single, closed region, and justify your choice. (c) Write a definite integral that gives the volume of the solid generated by revolving the region from part (b) about y=−2y=-2. (d) Evaluate the integral in part (c) to obtain the exact volume.

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