Mathematics

Mathematics

Problem

The circle r=3r=3 completely contains the lemniscate r=3cos⁡2θr=3\cos 2\theta. Which of the following integrals gives the exact area of the lemniscate? a) 2∫0π/412(3cos⁡2θ)2 dθ\displaystyle 2\int_{0}^{\pi/4} \tfrac12\bigl(3\cos 2\theta\bigr)^2\,d\theta b) 4∫0π/412(3cos⁡2θ)2 dθ\displaystyle 4\int_{0}^{\pi/4} \tfrac12\bigl(3\cos 2\theta\bigr)^2\,d\theta c) 4∫0π/212(3cos⁡2θ)2 dθ\displaystyle 4\int_{0}^{\pi/2} \tfrac12\bigl(3\cos 2\theta\bigr)^2\,d\theta d) 4∫0π/4(3cos⁡2θ)2 dθ\displaystyle 4\int_{0}^{\pi/4} \bigl(3\cos 2\theta\bigr)^2\,d\theta

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