Mathematics

Mathematics

Problem

The curve defined by r=1+cos⁡θr = 1 + \cos \theta is a cardioid. Which of the following definite integrals equals the total area enclosed by the curve? a) ∫02π(1+cos⁡θ) dθ\displaystyle \int_{0}^{2\pi}(1+\cos\theta)\,d\theta b) 12∫02π(1+cos⁡θ)2 dθ\displaystyle \tfrac12\int_{0}^{2\pi}(1+\cos\theta)^2\,d\theta c) ∫02π(1+cos⁡θ)2 dθ\displaystyle \int_{0}^{2\pi}(1+\cos\theta)^2\,d\theta d) 12∫0π(1+cos⁡θ)2 dθ\displaystyle \tfrac12\int_{0}^{\pi}(1+\cos\theta)^2\,d\theta

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