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Consider the cardioid in polar form $r=a(1-\cos\theta)$ (the standard “cycloidal” cusp curve used in calculus problems). Its total length is obtained via the polar arc-length formula $$ L=\int_{0}^{2\pi}\sqrt{r^{2}+\Big(\frac{dr}{d\theta}\Big)^{2}}\,d\theta,\qquad r=a(1-\cos\theta),\ \ r_\theta=a\sin\theta. ________ __________ ___ _______ _________.
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