Mathematics

Mathematics

Problem

On 0≤θ≤π20\le\theta\le\tfrac{\pi}{2}, the cardioid r=2+2cos⁡θr=2+2\cos\theta lies outside the circle r=2r=2. Which of the following integrals represents the area of the region between the two curves on this interval? a) ∫0π/2[(2+2cos⁡θ)2−22] dθ\displaystyle \int_{0}^{\pi/2}\bigl[(2+2\cos\theta)^2-2^2\bigr]\,d\theta b) 12∫0π/2[(2+2cos⁡θ)−2]2 dθ\displaystyle \tfrac12\int_{0}^{\pi/2}\bigl[(2+2\cos\theta)-2\bigr]^2\,d\theta c) 12∫0π/2[(2+2cos⁡θ)2−22] dθ\displaystyle \tfrac12\int_{0}^{\pi/2}\bigl[(2+2\cos\theta)^2-2^2\bigr]\,d\theta d) 12∫0π[(2+2cos⁡θ)2−22] dθ\displaystyle \tfrac12\int_{0}^{\pi}\bigl[(2+2\cos\theta)^2-2^2\bigr]\,d\theta

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