Which of the following integrals represents the total arc length of the cardioid…

MATHEMATICS · Single correct · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

Which of the following integrals represents the total arc length of the cardioid defined by r=1+cos⁡θr=1+\cos\theta as θ\theta traces the curve exactly once? (a) ∫02π(1+cos⁡θ)2+cos⁡2θ dθ\displaystyle \int_{0}^{2\pi}\sqrt{(1+\cos\theta)^{2}+\cos^{2}\theta}\,d\theta (b) ∫02π(1+cos⁡θ)2+sin⁡2θ dθ\displaystyle \int_{0}^{2\pi}\sqrt{(1+\cos\theta)^{2}+\sin^{2}\theta}\,d\theta (c) ∫0π(1+cos⁡θ)2+sin⁡2θ dθ\displaystyle \int_{0}^{\pi}\sqrt{(1+\cos\theta)^{2}+\sin^{2}\theta}\,d\theta (d) ∫02π(1+cos⁡θ) dθ\displaystyle \int_{0}^{2\pi}(1+\cos\theta)\,d\theta

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